Let \(\mathcal{G}\) be a family of graphs. The anti-Ramsey number \(\text{AR}(n, \mathcal{G})\) for \(\mathcal{G}\) is the maximum number
of colors in an edge coloring of \(K_n\) that has no rainbow copy of
any graph in \(\mathcal{G}\). In this paper, we determine the bipartite anti-Ramsey number for the family of trees with
\(k\) edges.

Let \(G\) be a finite group of order \(n\) and \(S\) (possibly containing the identity element) be a subset of \(G\). The Bi-Cayley graph
\(\text{BC}(G, S)\) of \(G\) is a bipartite graph with vertex set \(G \times \{0, 1\}\) and edge set \(\{(g, 0), (gs, 1) \mid g \in G, s \in S\}\). Let \(p\) (\(0 < p < 1\)) be a fixed number.We define \({B} = \{\text{BC}(G, S) \mid S \subseteq G\}\) as a sample space and assign a probability measure by requiring \(P_r(X) = p^k q^{n-k}\) for \(X = \text{BC}(G, S)\) with \(|S| = k\), where \(q = 1-p\). It is shown that the probability of the set of Bi-Cayley graphs of \(G\) with diameter \(3\) approaches \(1\) as the order \(n\) of \(G\) approaches infinity.

In this study, we define and investigate the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. We derive generating functions, Binet formulas, explicit formulas, and matrix representations for these numbers. Additionally, we present explicit combinatorial and determinantal expressions, examine negatively subscripted numbers, and establish various identities. Our results parallel those for the Jacobsthal and Jacobsthal Lucas numbers, yielding interesting consequences for the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers.

A signed total \(k\)-dominating function of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \{+1, -1\}\) such that for every vertex \(v\), the sum of the values of \(f\) over the open neighborhood of \(v\) is at least \(k\). A signed total \(k\)-dominating function \(f\) is minimal if there does not exist a signed total \(k\)-dominating function \(g\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\).The weight of a signed total \(k\)-dominating function is \(w(f) = \sum_{v \in V} f(v)\). The signed total \(k\)-domination number of \(G\), denoted by \(\gamma_{t,k}^s(G)\), is the minimum weight of a signed total \(k\)-dominating function on \(G\).The upper signed total \(k\)-domination number \(\Gamma_{t,k}^s(G)\) of \(G\) is the maximum weight of a minimal signed total \(k\)-dominating function on \(G\).
In this paper, we present sharp lower bounds on \(\gamma_{t,k}^s(G)\) for general graphs and \(K_{r+1}\)-free graphs and characterize the extremal graphs attaining some lower bounds. Also, we give a sharp upper bound on \(\Gamma_{t,k}^s(G)\) for an arbitrary graph.

We show that a \(2\)-subset-regular self-complementary \(3\)-uniform hypergraph with \(7\) vertices exists if and only if \(n \geq 6\) and \(n\) is congruent to \(2\) modulo \(4\).

Given a graph \(G\), a function \(f: V(G) \to \{1, 2, \ldots, k\}\) is a \(k\)-ranking of \(G\) if \(f(u) = f(v)\) implies every \(u-v\)
path contains a vertex \(w\) such that \(f(w) > f(u)\). A \(k\)-ranking is minimal if the reduction of any label greater
than \(1\) violates the described ranking property.The \(arank\) number of a graph, denoted \(\psi_r(G)\),
is the maximum \(k\) such that \(G\) has a minimal \(k\)-ranking.We establish new properties for minimal rankings and present
new results for the \(arank\) number of a cycle.

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs \(G_1\) and \(G_2\) are equal to \(\kappa_1 v_2\) and \(\min\{\lambda_1 v_2^2, \delta_2 + \delta_1v_2\}\), respectively, where \(\delta_i\), \(\kappa_i\), \(\lambda_i\), and \(n_i\) denote the minimum degree, connectivity, edge-connectivity, and number of vertices of \(G_i\), respectively.
We also obtain that the edge-connectivity of the direct product of \(K_2\) and a graph \(H\) is equal to \(\min\{2\lambda, 2\beta, \min_{j =\lambda}^\delta\{j + 2\beta_j\}\}\), where \(\theta\) is the minimum size of a subset \(F \subset E(H)\) such that \(H – F\) is bipartite and \(\beta_j = \min\{\beta(C)\}\), where \(C\) takes over all components of \(H – B\) for all edge-cuts \(B\) of size \(j \geq \lambda=\lambda (H)\).

Consider an n-set, say \(X_n = {1,2,…,n}\). An exponential generating function and recurrence relation for the number of subpermutations of \(X_n\), whose orbits are of size at most \(k \geq 0\) are obtained. Similar results for
the number of nilpotent subpermutations of nilpotency index at most \(k\), and exactly \k\) are also given, along with arithmetic and asypmtotic formulas for these numbers. \(1\) \(2\)

In this paper, we show that the crossing number of the complete tripartite graph \(K_{2,4,n}\) is \(6\left\lfloor\frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor+2n\).

An \((n \times n)\) matrix \(A = (a_{ij})\) is called a Toeplitz matrix
if it has constant values along all diagonals parallel to the main diagonal.
A directed Toeplitz graph is a digraph with Toeplitz adjacency matrix.
In this paper, we discuss conditions for the existence of Hamiltonian cycles
in directed Toeplitz graphs.

For \(n \geq 2\) and a local field \(K\), let \(\Delta_n\) denote the affine building naturally associated to the symplectic group \(\mathrm{Sp}_{n}(K)\). We compute the spectral radius of the subgraph \(Y_n\) of \(\Delta_n\) induced by the special vertices in \(\Delta_n\), from which it follows that \(Y_n\) is an analogue of a family of expanders and is non-amenable.

The concept of \(t\)-(v, \(\lambda\)) trades of block designs has been studied in detail. See, for example, A.~S. Hedayat (1990) and Billington (2003). Latin trades have also been extensively studied under various names; see A.~D. Keedwell (2004) for a survey. Recently, Khanban, Mahdian, and Mahmoodian have extended the concept of Latin trades and introduced \(t\)-(\(v, k\)) Latin trades.In this paper, we study the spectrum of possible volumes of these trades, \(S(t, k)\). Firstly, similarly to trades of block designs, we consider \((t+2)\) numbers \(s_i = 2^{i+1}-2^{(t+1)-i} \), \(0 \leq i \leq t+1\), as critical points. Then, we show that \(s_i \in S(t,k)\) for any \(0 \leq i \leq t+1\), and if \(s \in (s_i, s_{i+1}, )\), \(0 \leq i \leq t\), then \(s \notin S(t, t+1)\). As an example, we precisely determine \(S(3, 4)\).

This paper investigates the relationship between the degree-sum of adjacent vertices, girth, and upper embeddability of graphs, combining it with edge-connectivity. The main result is:
Let \(G\) be a \(k\)-edge-connected simple graph with girth \(g\). If there exists an integer \(m\) (\(1 \leq m \leq g\)) such that for any \(m\) consecutively adjacent vertices \(x_i\) (\(i = 1, 2, \ldots, m\)) in any non-chord cycle \(C\) of \(G\), it holds that

\[\sum\limits_{i=1}^m d_G(x_i) > \frac{mn}{(k-1)^2+2} + \frac{km}{g}+(2-g)m,\]

where \(k = 1, 2, 3, n = |V(G)|\), then \(G\) is upper embeddable and the upper bound is best possible.

In this study, we define and investigate the Bivariate Gaussian Fibonacci and Bivariate Gaussian Lucas Polynomials. We derive generating functions, Binet formulas, explicit formulas, and partial derivatives of these polynomials. By defining these bivariate polynomials for special cases, we obtain:\(F_n(x, 1)\) as the Gaussian Fibonacci polynomials,\(L_n(x, 1)\) is the Gaussian Lucas polynomials,\( {F}_{n}(1, 1)\) as the Gaussian Fibonacci numbers, and \( {L}_{n}(1, 1)\) as the Gaussian Lucas numbers, as defined in \([19]\).

In this paper, we show that the set \(\{E_0(x), E_1(x), \ldots, E_n(x)\}\) of Euler polynomials is a basis for the space of polynomials of degree less than or equal to \(n\). From the properties of Euler basis polynomials, we derive some interesting identities on the product of two Bernoulli and Euler polynomials.

An \(n\)-colour even composition is defined as an \(n\)-colour composition with even parts. In this paper, we obtain generating functions, explicit formulas, and a recurrence formula for \(n\)-colour even compositions.

In this paper, we characterize boundedness and compactness of products of composition operators induced by the lens and the lunar maps and iterated differentiation acting between Hardy and weighted Bergman spaces of the unit disk in terms of the angle of contact of these maps with the unit circle.

Let \(G = (V(G), E(G))\) be a graph and \(\alpha(G)\) be the independence number of \(G\). For a vertex \(v \in V(G)\), \(d(v)\) and \(N(v)\) represent the degree and the neighborhood of \(v\) in \(G\), respectively.In this paper, we prove that if \(G\) is a \(k\)-connected graph of order \(n\), where (\(k \geq 2\)) graph of order \(n\) and \(\max\{d(v) : v \in S\} \geq \frac{n}{2}\) for every independent set \(S\) of \(G\) with \(|S| = k\) which has two distinct vertices \(x, y \in S\) satisfying \(1\leq |N(x) \cap N(y)| \leq \alpha(G) – 2,\)
then either \(G\) is hamiltonian or else \(G\) belongs to one of a family of exceptional graphs.We also establish a similar sufficient condition for Hamiltonian-connected graphs.

In this paper, we generalize the companion Pell sequence. We provide combinatorial, graph, and matrix representations of this sequence.Using these representations, we describe some properties of the generalized Pell numbers and the generalized companion Pell numbers. We define the golden Pell matrix for determining the generalized Pell sequences and, among other results, prove the “generalized Cassini formula” for them.Moreover, we establish some relations between generalized Pell numbers and the classical Fibonacci numbers.

In this paper, we determine the third largest and the fourth largest numbers of independent sets among all trees of order \(n\). Moreover, we determine the \(k\)-th largest numbers of independent sets among all forests of order \(n\), where \(k \geq 2\). Besides, we characterize those extremal graphs achieving these values.

For a set \(\mathcal{P}\) of permutations, the sign-imbalance of \(\mathcal{P}\) is the difference between the numbers of even and odd permutations in \(\mathcal{P}\).In this paper, we determine the sign-imbalances of two classes of alternating permutations ,one is the Alternating permutations avoiding a pattern of length three and the other is the Alternating permutations of genus \(0\)
The sign-imbalance of the former involves Catalan and Fine numbers, and that of the latter is always \(\pm 1\).Meanwhile, we give a simpler proof of Dulucq and Simion’s result on the number of alternating permutations of genus \(0\).

A survivable path \((W, P)\) between a pair of vertices \(x_i, x_j\) in an undirected simple graph \(G\) is an ordered pair of edge-disjoint simple paths consisting of a working path \(W = x_i, \ldots, x_j\) a protection path \(P = x_i, \ldots, x_j\).An optimal set of survivable paths in graph \(G\) corresponds to a set of mesh-restored lightpaths defined on an optical network that minimizes the number of used optical channels.In this paper, we present new properties of the working paths, which are contained in an optimal set of survivable paths in \(G\).

We describe the global behavior of the nonnegative equilibrium points of the difference equation

\[x_{n+1} = \frac{ax_{n -p}}{b+c \prod\limits_{i=0}^{k} x_{n-(2i+1)}},n=0,1,\ldots,\]

where \(k,p \in \mathbb{N}\), parameters \(a,b,c\) and initial conditions are nonnegative real numbers.

Let \(\mathcal{T}_{n,n-4}\) be the set of trees on \(n\) vertices with diameter \(n-4\). In this paper, we determine the unique tree which has the minimal Laplacian spectral radius among all trees in \(\mathcal{T}_{n,n-4}\).
This work is related to that of Yuan [The minimal spectral radius of graphs of order n with diameter \(n – 4\), Linear Algebra Appl. \(428(2008)2840-2851]\), which determined the graph with minimal spectral radius among all the graphs of order \(n\) with diameter \(n-4\). We can observe that the extremal tree on the Laplacian spectral radius is different from that on the spectral radius.

We introduce the notion of vague Lie sub-superalgebras (resp. vague ideals) and present some of their properties. We investigate the properties of vague Lie sub-superalgebras and vague ideals under homomorphisms of Lie superalgebras.We introduce the concept of vague bracket product and establish its characterizations. We also introduce the notions of solvable vague ideals and nilpotent vague ideals of Lie superalgebras and present the corresponding theorems parallel to Lie superalgebras.

The atom-bond connectivity (ABC) index of a graph \(G\) is defined in mathematical chemistry as\(\mathrm{ABC}(G) = \sum_{uv \in E(G)} \sqrt{\frac{d_u +d_v-2}{ d_u d_v}},\) where \(E(G)\) is the edge set of \(G\) and \(d_u\) is the degree of vertex \(u\) in \(G\).In this paper, we determine the unique graphs with the largest and the second largest ABC indices, respectively, in the class of unicyclic graphs on \(2m\) vertices with perfect matchings.

Let \(\Delta\) be one of the dual polar spaces \(\mathrm{DQ}(8, q)\), \(\mathrm{DQ}^-(7,q)\), and let \(e: \Delta \to \Sigma\) denote the spin-embedding of \(\Delta\). We show that \(e(\Delta)\) is a two-intersection set of the projective space \(\Sigma\). Moreover, if \(\Delta \cong \mathrm{DQ}^-(7,q)\), then \(e(\Delta)\) is a \((q^3 + 1)\)-tight set of a nonsingular hyperbolic quadric \(\mathrm{Q}^+(7,q^2)\) of \(\Sigma \cong PG(7,q^2)\). This \((q^2 + 1)\)-tight set gives rise to more examples of \((q^3 + 1)\)-tight sets of hyperbolic quadrics by a procedure called field-reduction.All the above examples of two-intersection sets and \((q^3 + 1)\)-tight sets give rise to two-weight codes and strongly regular graphs.

Let \(G = (V, E)\) be a simple undirected graph. An independent set is a subset \(S \subseteq V\) such that no two vertices in \(S\) are adjacent. A maximal independent set is an independent set that is not a proper subset of any other independent set.
In this paper, we study the problem of determining the fourth largest number of maximal independent sets among all trees and forests. Extremal graphs achieving these values are also given.

From differential operators and the generating functions of Bernoulli and Euler polynomials, we derive some new theorems on Bernoulli and Euler numbers. By using integral formulae and arithmetical properties relating to the Bernoulli and Euler polynomials, we obtain new identities on Bernoulli and Euler numbers. Finally, we give some new properties on Bernoulli and Euler numbers arising from the \(p\)-adic integrals on \(\mathbb{Z}_p\).

Let \(u,v\) be two vertices of a connected graph \(G\). The vertex \(v\) is said to be a boundary vertex of \(u\) if no neighbor of \(v\) is further away from \(u\) than \(v\). The boundary of a graph is the set of all its boundary vertices.In this work, we present a number of properties of the boundary of a graph under different points of view:(1) A realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary.(2) The contour is a monophonic set.(3) The cardinality of the boundary is an upper bound for both the metric dimension and the determining number of a graph.

Computing the crossing number of a given graph is, in general, an elusive problem, and only the crossing numbers of a few families of graphs are known. Most of them are the Cartesian products of special graphs. This paper determines the crossing number of the Cartesian product of a 6-vertex graph with the star \(S_n\).

Let \(M = (E, \mathcal{F})\) be a matroid on a set \(E\), \(B\) one of its bases, and \(M_B\) the base matroid associated to \(B\). In this paper, we determine a characterization of simple binary matroids \(M\) which are not isomorphic to \(M_B\), for every base \(B\) of \(M\). We also extend to matroids some graph notions.

Let \(H\) and \(G\) be two graphs (or digraphs), where \(G\) is a subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \((H,G)\)-GD, is a partition of all the edges (or arcs) of \(H\) into subgraphs (\(G\)-blocks), each of which is isomorphic to \(G\). A large set of \((H, G)\)-GD, denoted by \((H, G)\)-LGD, is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \((H,G)\)-GDs. In this paper, we obtain the existence spectra of \((ADK_{m,n}, P_3^i)\)-LGD, where \(P_3^i\) (\(i = 1,2,3\)) are the three types of oriented \(P_3\).

Let \(G\) be a graph. The zeroth-order general Randić index of a graph is defined as \(R_\alpha^0(G) = \sum_{v \in V(G)} d(v)^\alpha(v)\), where \(\alpha\) is an arbitrary real number and \(d(v)\) is the degree of the vertex \(v\) in \(G\). In this paper, we give sharp lower and upper bounds for the zeroth-order general Randić index \(R_\alpha^0(G)\) among all unicycle graphs \(G\) with \(n\) vertices and \(k\) pendant vertices.

\(n\)-ary hypergroups are a generalization of Dörnte \(n\)-ary groups and a generalization of hypergroups in the sense of Marty. In this paper, we investigate some properties of \(n\)-ary hypergroups and (commutative) fundamental relations. We determine two families \( {P}(H)\) and \( {P}_\sigma(H)\) of subsets of an \(n\)-ary hypergroup \(H\) such that two geometric spaces \((H, {P}(H))\) and \((H, {P}_\sigma(H))\) are strongly transitive. We prove that in every \(n\)-ary hypergroup, the fundamental relation \(\beta\) and the commutative fundamental relation \(\gamma\) are strongly compatible equivalence relations.

In this paper, we develop a technique that allows us to obtain new effective constructions of \(1\)-resilient Boolean functions with very good nonlinearity and autocorrelation. Our strategy to construct a \(1\)-resilient function is based on modifying a bent function by toggling some of its output bits. Two natural questions that arise in this context are: “At least how many bits and which bits in the output of a bent function need to be changed to construct a \(1\)-resilient Boolean function?” We present an algorithm that determines a minimum number of bits of a bent function that need to be changed to construct a \(1\)-resilient Boolean function. We also present a technique to compute points whose output in the bent function need to be modified to get a \(1\)-resilient function. In particular, the technique is applied up to \(14\)-variable functions, and we show that the construction provides \(1\)-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation absolute indicator values, which were not known earlier.

The noncrossing matchings with each of their blocks containing a given element are introduced and studied. The enumeration of these matchings is described through a polynomial of several variables, which is proved to satisfy a recursive formula. Results of the enumeration of noncrossing matchings with fixed points are connected with Catalan numbers.

For \(1 \leq s \leq n-3\), let \(C_n(i;i_1, v_2, \ldots, i_s)\) denote an \(n\)-cycle with consecutive vertices \(x_1, x_2, \ldots, x_n\) to which the \(s\) chords \(x_{ i}x_{i_1}, x_{i}x_{i_2}, \ldots, x_{i}x_{i_s}\) have been added. In this paper, we discuss the strongly \(c\)-harmonious problem of the graph \(C_n(i;i_1, i_2, \ldots, i_s)\).

A shell of width \(n\) is a fan \(C_n(1;3,4, \ldots, n-1)\) and a vertex with degree \(n-1\) is called apex. \(MS(n^m)\) is a graph consisting of \(m\) copies of shell of width \(n\) having a common apex. If \(m \geq 1\) is odd, then the multiple shell \(MS(n^ m)\) is harmonious.

The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\) is the set consisting of \(ev\) and of all edges having a common end-vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1, 1\}\). If \(\sum_{x\in E(G)}f(x \geq 1\) for at least \(k\) edges \(e\) of \(G\), then \(f\) is called a signed edge \(k\)-subdominating function of \(G\). The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over all signed edge \(k\)-subdominating functions \(f\) of \(G\), is called the signed edge \(k\)-subdomination number of \(G\) and is denoted by \(\gamma_{s,k}(G)\). In this note, we initiate the study of the signed edge \(k\)-subdomination in graphs and present some (sharp) bounds for this parameter.

A graph \(G\) with vertex set \(V\) is said to have a prime labeling if its vertices can be labeled with distinct integers \(1, 2, \ldots, |V|\) such that for every edge \(xy\) in \(E(G)\), the labels assigned to \(x\) and \(y\) are relatively prime or coprime. In this paper, we show that the Knödel graph \(W_{3,n}\) is prime for \(n \leq 130\).

Let \(G = (V, E)\) be a graph. A set \(D \subseteq V\) is a total restrained dominating set of \(G\) if every vertex in \(V\) has a neighbor in \(D\) and every vertex in \(V – D\) has a neighbor in \(V – D\). The cardinality of a minimum total restrained dominating set in \(G\) is the total restrained domination number of \(G\). In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for the total restrained domination number of graphs, and constructively characterize trees having their total restrained domination numbers achieving the lower bound.

Let \(\Gamma = (X, R)\) denote a \(d\)-bounded distance-regular graph with diameter \(d \geq 3\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). For \(0 \leq i \leq i+s \leq d-1\), suppose \(\Delta_i\) and \(\Delta_0\) are subspaces with diameter \(i\) and \(i+s\), respectively, and with \(\Delta_i \subseteq \Delta_0\). Let \(\mathcal{L}(i, i+s; d)\) denote the set of all subspaces \(\Delta’\) with diameters \(\geq i\) such that \(d(\Delta_0 \cap \Delta’) = \Delta_1\) and \(d(\Delta_0 + \Delta’) = d(\Delta’) + s\) in \(\Gamma$ including \(\Delta_0\). If we partial order \(\mathcal{L}(i, i+s; d)\) by ordinary inclusion (resp. reverse inclusion), then \(\mathcal{L}(i, i+s; d)\) is a poset, denoted by \(\mathcal{L}_0(i, i+s; d)\) (resp. \(\mathcal{L}_R(i, i+s; d)\)). In the present paper, we show that both \(\mathcal{L}_0(i, i+s; d)\) and \(\mathcal{L}_R(i, i+s; d)\) are atomic lattices, and classify their geometricity.

By means of the partial fraction decomposition method, we evaluate a very general determinant of formal shifted factorial fractions, which covers numerous binomial determinantal identities.

Let \(H\) be a subgroup of a finite group \(G\). The relative \(n\)-th commutativity degree, denoted as \(P_n(H,G)\), is the probability of commuting the \(n\)-th power of a random element of \(H\) with an element of \(G\). Obviously, if \(H = G\) then the relative \(n\)-th commutativity degree coincides with the \(n\)-th commutativity degree, \(P_n(G)\). The purpose of this article is to compute the explicit formula for \(P_n(G)\), where \(G\) is a 2-generator \(p\)-group of nilpotency class two. Furthermore, we observe that if we have two pairs of relative isoclinic groups, then they have equal relative \(n\)-th commutativity degree.

Let \(\varphi: M \to {C}^n\) be an \(n\)-dimensional compact Willmore Lagrangian submanifold in the Complex Euclidean Space \({C}^n\). Denote by \(S\) and \(H\) the square of the length of the second fundamental form and the mean curvature of \(M\), respectively. Let \(p\) be the non-negative function on \(M\) defined by \(p^2 = S – nH^2\). Let \(K\) and \(Q\) be the functions which assign to each point of \(M\) the infimum of the sectional curvature and Ricci curvature at the point, respectively. In this paper, we prove some integral inequalities of Simons’ type for \(n\)-dimensional compact Willmore Lagrangian submanifolds \(\varphi: M \to {C}^n\) in the Complex Euclidean Space \({C}^n\) in terms of \(p^2\), \(K\), \(Q\), and \(H\), and give some rigidity and characterization theorems.

Let \(G\) be a subgraph of \(K_n\). The graph obtained from \(G\) by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a \(T(G)\)-triple. An edge-disjoint decomposition of \(3K_n\) into copies of \(T(G)\) is called a \(T(G)\)-triple system of order \(n\). If, in each copy of \(T(G)\) in a \(T(G)\)-triple system, one edge is taken from each 3-cycle (chosen so that these edges form a copy of \(G\)) in such a way that the resulting copies of \(G\) form an edge-disjoint decomposition of \(K_n\), then the \(T(G)\)-triple system is said to be perfect. The set of positive integers \(n\) for which a perfect \(T(G)\)-triple system exists is called its spectrum. Earlier papers by authors including Billington, Lindner, Kıygıkçifi, and Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). Then, in our previous paper, the spectrum of perfect \(T(G)\)-triple systems for each graph \(G\) with five vertices and \(i (\leq 6)\) edges was determined. In this paper, we will completely solve the spectrum problem of perfect \(T(G)\)-triple systems for each graph \(G\) with five vertices and seven edges.

This paper investigates tilings of a \(2 \times n\) rectangle using vertical and horizontal dominos. It is well-known that these tilings are counted by the Fibonacci numbers. We associate a graph to each tiling by converting the corners and borders of the dominos to vertices and edges. We study the combinatorial, probabilistic, and graph-theoretic properties of the resulting “domino tiling graphs.” In particular, we prove central limit theorems for naturally occurring statistics on these graphs. Some of these results are then extended to more general tiling graphs.

We recall from [13] a shell graph of size \(n\), denoted \(C(n, n-3)\), is the graph obtained from the cycle \(C_n(v_1, v_2, \ldots, v_{n-1})\) by adding \(n-3\) consecutive chords incident at a common vertex, say \(v_0\). The vertex \(v_0\) of \(C(n, n-3)\) is called the apex of the shell \(C(n, n-3)\). The vertex \(v_1\) of \(C(n, n-3)\) is said to be at level 1.

A graph \(C(2n,n-2)\) is called an alternate shell, if \(C(2n,n-2)\) is obtained from the cycle \(C_{2n}(v_0,v_1, v_2, \ldots, v_{2n-1})\) by adding \(n-2\) chords between the vertex \(v_0\) and the vertices \(v_{2i+1}\), for \(1\leq i \leq n-2\). If the vertex \(v_i\) of \(C(2n,n-2)\) at level 1 is adjacent with \(v_0\), then \(v_1\) is said to be at level 1 with a chord, otherwise the vertex \(v_1\) is said to be at level 1 without a chord.

In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph \(D\), which is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex \(w\) such that we can get to \(w\) from \(u\) and \(v\) in \(D\) by walks of length \(k\). In this paper, we study and obtain the scrambling indices of all primitive digraphs with exactly two cycles.

Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for every \(a, b \in X\) and \(x \in V – X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament \(T\) of cardinality \(\geq 5\) such that for any vertex \(x\) of \(T\), the tournament \(T – x\) is decomposable. The critical tournaments are of odd cardinality and for all \(n \geq 2\) there are exactly three critical tournaments on \(2n + 1\) vertices denoted by \(T_{2n+1}\), \(U_{2n+1}\), and \(W_{2n+1}\). The tournaments \(T_5\), \(U_5\), and \(W_5\) are the unique indecomposable tournaments on 5 vertices. We say that a tournament \(T\) embeds into a tournament \(T’\) when \(T\) is isomorphic to a subtournament of \(T’\). A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and \(T_5\) embed into an indecomposable tournament \(T\), then \(W_5\) and \(U_5\) embed into \(T’\). To conclude, we prove the following: given an indecomposable tournament \(T\) with \(|V(T)| \geq 7\), \(T\) is critical if and only if only one of the tournaments \(T_7\), \(U_7\), or \(W_7\) embeds into \(T\).

Let \(\lambda K_{m,n}\) be a complete bipartite multigraph with two partite sets having \(m\) and \(n\) vertices, respectively. A \(K_{p,q}\)-factorization of \(\lambda K_{m,n}\) is a set of edge-disjoint \(K_{p,q}\)-factors of \(\lambda K_{m,n}\) which is a partition of the set of edges of \(\lambda K_{m,n}\). When \(\lambda = 1\), Martin, in paper [Complete bipartite factorisations by complete bipartite graphs, Discrete Math., \(167/168 (1997), 461-480]\), gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will give similar necessary conditions for \(\lambda K_{m,n}\) to have a \(K_{p,q}\)-factorization, and prove the necessary conditions are always sufficient in many cases.

In this paper, we determine upper and lower bounds for the number of independent sets in a bicyclic graph in terms of its order. This
gives an upper bound for the total number of independent sets in a connected graph which contains at least two cycles. In each case, we characterize the extremal graphs.

Let \(G\) be a connected graph of order \(n\). Denote \(p_u(G)\) the order of a longest path starting at vertex \(u\) in \(G\). In this paper, we prove that if \(G\) has more than \(t\binom{k}{2} + \binom{p+1}{2} + (n-k-1)\) edges, where \(k \geq 2\), \(n = t(k-1) + p + 1\), \(t \geq 0\) and \(0 \leq p \leq k-1\), then \(p_u(G) > k\) for each vertex \(u\) in \(G\). By this result, we give an alternative proof of a result obtained by P. Wang et al. that if \(G\) is a 2-connected graph on \(n\) vertices and with more than \(t\binom{k-2}{2} + \binom{p}{2} + (2n – 3)\) edges, where \(k \geq 3\), \(n-2 = t(k-2) + p\), \(t \geq 0\) and \(0 \leq p \leq k-2\), then each edge of \(G\) lies on a cycle of order more than \(k\).

In this paper, we give some identities involving the harmonic numbers and the inverses of binomial coefficients.

In this paper, a new efficient computational algorithm is presented for solving cyclic heptadiagonal linear systems based on using the heptadiagonal linear solver and Sherman–Morrison–Woodbury formula. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Two numerical examples are presented for illustration.

Let \(G\) be a graph, and let \(a, b\), and \(k\) be nonnegative integers with \(0 \leq a \leq b\). A graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\), the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, we prove that if \(\delta(G) \geq a + k\) and \(\alpha(G) \leq \frac{4b(\delta(G)-a+1-1)}{(a+1)^2}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.

A characterization of \(B\)-H-unretractive bipartite graphs is given. Based on this, it is proved that there is no bipartite graph with endotype \(1 \pmod{4}\).

In a graph \(G = (V, E)\), an independent set is a subset \(I\) of \(V(G)\) such that no two vertices in \(I\) are adjacent. A maximum independent set is an independent set of maximum size. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we study the problem of determining the largest and the second largest numbers of maximum independent sets among all quasi-tree graphs and quasi-forest graphs. Extremal graphs achieving these values are also given.

The notions of sum labelling and sum number of graphs were introduced by F. Harary [1] in 1990. A mapping \(f\) is called a sum labelling of a graph \(G(V, E)\) if it is an injection from \(V\) to a set of positive integers such that \(uv \in E\) if and only if there exists a vertex \(w \in V\) such that \(f(w) = f(x) + f(y)\). In this case, \(w\) is called a working vertex. If \(f\) is a sum labelling of \(G\) with \(r\) isolated vertices, for some nonnegative integer \(r\), and \(G\) contains no working vertex, \(f\) is defined as an exclusive sum labelling of the graph \(G\) by M. Miller et al. in paper [2]. The least possible number \(r\) of such isolated vertices is called the exclusive sum number of \(G\), denoted by \(\epsilon(G)\). If \(\epsilon(G) = \Delta(G)\), the labelling is called \(\Delta\)-optimum exclusive sum labelling and the graph is said to be \(\Delta\)-optimum summable, where \(\Delta = \Delta(G)\) denotes the maximum degree of vertices in \(G\). By using the notion of \(\Delta\)-optimum forbidden subgraph of a graph, the exclusive sum numbers of crown \(C_n \odot K_1\) and \((C_n \odot K_1)\) are given in this paper. Some \(\Delta\)-optimum forbidden subgraphs of trees are studied, and we prove that for any integer \(\Delta \geq 3\), there exist trees not \(\Delta\)-optimum summable. A nontrivial upper bound of the exclusive sum numbers of trees is also given in this paper.

In this paper we obtain the Fibonacci length of amalgamated free products having as factors dihedral groups.

In [11], Zhu, Li, and Deng introduced the definition of implicit degree of a vertex \(v\), denoted by \(\text{id}(v)\). In this paper, we consider implicit degrees and the hamiltonicity of graphs and obtain that:
If \(G\) is a \(2\)-connected graph of order \(n\) such that \(\text{id}(u) + \text{id}(v) \geq n – 1\) for each pair of vertices \(u\) and \(v\) at distance \(2\), then \(G\) is hamiltonian, with some exceptions.

Let \(C_k\) denote a cycle of length \(k\) and let \(S_k\) denote a star with \(k\) edges. For graphs \(F\), \(G\), and \(H\), a \((G, H)\)-multidecomposition of \(F\) is a partition of the edge set of \(F\) into copies of \(G\) and copies of \(H\) with at least one copy of \(G\) and at least one copy of \(H\). In this paper, necessary and sufficient conditions for the existence of the \((C_k, S_k)\)-multidecomposition of a complete bipartite graph are given.

This paper investigates the number of boundary cubic inner-forest maps and presents some formulae for such maps with the size (number of edges) and the valency of the root-face as two parameters. Further, by duality, some corresponding results for rooted outer-planar maps are obtained. It is also an answer to the open problem in \([15]\) and corrects the result on boundary cubic inner-tree maps in \([15]\).

The following two theorems are proved:
A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a cylinder so that the \(m\) rows go around the cylinder, with one square removed, with the exception of the following boards:

(a) \(n\) is even,

(b) \(m \in \{1,2\}\)

(c) \(m = 4\) and the removed square is in row 2 or 3;

(d) \(m \geq 5\), \(n = 1\), and the removed square is in row 2, 3, …, or \(m-1\).

A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a torus with one square removed except boards with \(m\) and \(n\) both even and \(1 \times 1\),\(1 \times 2\) and \(2 \times 1\) boards.

An independent set \(S\) of a connected graph \(G\) is called a \emph{frame} if \(G – S\) is connected. If \(|S| = k\), then \(S\) is called a \emph{k-frame}. We prove the following theorem.
Let \(k \geq 2\) be an integer, \(G\) be a connected graph with \(V(G) = \{v_1, v_2, \ldots, v_n\}\), and \(\deg_G(u)\) denote the degree of a vertex \(u\). Suppose that for every \(3\)-frame \(S = \{v_a, v_b, v_c\}\) such that \(1 \leq a \leq b \leq c \leq n\), \(\deg_G(v_c) \leq a\), \(\deg_G(v_b) \leq b-1\), and \(\deg_G(v_c) \leq c – 2\), it holds that\[\deg_G(v_a) + \deg_G(v_b) + \deg_G(v_c) – |N(v_a) \cap N(v_b) \cap N(v_c)| \geq |G| – k + 1.\] Then \(G\) has a spanning tree with at most \(k\)-leaves. Moreover, the condition is sharp.
This theorem is a generalization of the results of E. Flandrin, H.A. Jung, and H. Li (Discrete Math. \(90 (1991), 41-52)\) and of A. Kyaw (Australasian Journal of Combinatorics. \(37 (2007), 3-10)\) for traceability.

In this paper, the estimations of maximum genus orientable embeddings of graphs are studied, and an exponential lower bound for such numbers is found. Moreover, such two extremal embeddings (i.e., the maximum genus orientable embedding of the current graph and the minimum genus orientable embedding of the complete graph) are sometimes closely related to each other. As applications, we estimate the number of minimum genus orientable embeddings for the complete graph by estimating the number of maximum genus orientable embeddings for the current graph.

In this article, we characterize for which finite commutative rings \(R\), The zero-divisor graph \(\Gamma(R)\),The line graph \(L(\Gamma(R))\), The complement graph \(\overline{\Gamma(R)}\), and The line graph for the complement graph \(L(\overline{\Gamma(R)})\).

The energy of a graph \(G\) is defined as the sum of the absolute values of all the eigenvalues of the graph. In this paper, we consider the energy of the \(3\)-circulant graphs, and obtain a computation formula, and establish new results for a certain class of circulant graphs. At the same time, we give a conjecture: The largest energy of circulant graphs relates with their components.

In this paper, we present two criteria for a sequence lying along a ray of a combinatorial triangle to be unimodal, and give a correct
proof for the result of Belbachir and Szalay on unimodal rays of the generalized Pascal’s triangle.

In this paper, we introduce the notion of derivation in lattice implication algebra, and consider the properties of derivations in lattice implication algebras. We give an equivalent condition to be a derivation of a lattice implication algebra. Also, we characterize the fixed set \(Fix_d(L)\) and \(Kerd\) by derivations. Moreover, we prove that if \(d\) is a derivation of a lattice implication algebra, every filter \(F\) is \(d\)-invariant.

In this paper, we derive a family of identities on the arbitrary subscripted Fibonacci and Lucas numbers. Furthermore, we construct the tridiagonal and symmetric tridiagonal family of matrices whose determinants form any linear subsequence of the Fibonacci numbers and Lucas numbers. Thus, we give a generalization of the results presented in Nalli and Civciv [A. Nalli, H. Civciv, A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos, Solitons and Fractals \(2009;40(1):355 .61]\) and Cahill and Narayan [N. D. Cahill, D. A. Narayan, Fibonacci and Lucas numbers as tridiagonal matrix determinants, The Fibonacci Quarterly, \(2004;42(1):216–221]\).

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we determine the second largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.

For a poset \(P = (X, \leq_ P)\), the strict-double-bound graph (\(sDB\)-graph \(sDB(P)\)) is the graph on \(X\) for which vertices \(u\) and \(v\) of \(sDB(P)\) are adjacent if and only if \(u \neq v\) and there exist \(x\) and \(y\) in \(X\) distinct from \(u\) and \(v\) such that \(x \leq_ P y\) and \(x \leq_P v \leq_P y\). The strict-double-bound number \(\zeta(G)\) of a graph \(G\) is defined as \(\min\{n; G \cup \overline{K}_n \text{ is a strict-double-bound graph}\}\).

We obtain that for a spider \(S_{n,m}\) (\(n,m > 3\)) and a ladder \(L_n\) (\(n \geq 4\)), \(\left\lceil2\sqrt{nm}\right\rceil \leq \zeta(S_{n,m}) \leq n+m\), \(\zeta(S_{n,n}) = 2n\), and \(\left\lceil 2\sqrt{3n+2}\right\rceil \leq \zeta(L_n) \leq 2n\).

We conjectured in \([3]\) that every biconnected cyclic graph is the one-dimensional skeleton of a regular cellulation of the \(3\)-sphere and proved it is true for planar and hamiltonian graphs. In this paper, we introduce the class of weakly split graphs and prove the conjecture is true for such class. Hamiltonian, split, complete \(k\)-partite, and matrogenic cyclic graphs are weakly split.

Let \((X,\mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition and let \(G_i\), \(i = 1,\ldots,\mu\), be nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i | B \in \mathcal{B}\}\), where \(\mathcal{B_i}\) is a subgraph of \(B\)  isomorphic to \(G_i\). A \(\{G_1,G_2,\ldots,G_\mu\}\)-metamorphosis of \((X,\mathcal{B})\) is a rearrangement, for each \(i=1,\ldots,\mu\), of the edges of \(\bigcup_{B\in B}(E(B)\setminus\mathcal{B}_i))\) into a family \(\mathcal{F}_i\) of copies of \(G_i\) with a leave \(L_i\), such that \((X,\mathcal{B}_i \cup \mathcal{F}_i,L_i)\) is a maximum packing of \(\lambda H\) with copies of \(G_i\). In this paper, we give a complete answer to the existence problem of an \(S_\lambda(2,4,7)\) having a \(\{C_4, K_3 + e\}\)-metamorphosis.

For a positive integer \(m\), where \(1 \leq m \leq n\), the \(m\)-competition index (generalized competition index) of a primitive digraph \(D\) of order \(n\) is the smallest positive integer \(k\) such that for every pair of vertices \(x\) and \(y\), there exist \(m\) distinct vertices \(v_1, v_2, \ldots, v_m\) such that there exist walks of length \(k\) from \(x\) to \(v_i\) and from \(y\) to \(v_i\) for \(1 \leq i \leq m\). In this paper, we study the generalized competition indices of symmetric primitive digraphs with loop. We determine the generalized competition index set and characterize completely the symmetric primitive digraphs in this class such that the generalized competition index is equal to the maximum value.

We give a new combinatorial bijection between a certain set of balanced modular tableaux of Gusein-Zade, Luengo, and Melle-Hernandez and \(k\)-ribbon shapes. In addition, we also use the Schensted algorithm for the rim hook tableaux of Stanton and White to write down an explicit generating function for these balanced modular tableaux.

A \((k;g)\)-cage is a graph with the minimum order among all \(k\)-regular graphs with girth \(g\). As a special family of graphs, \((k;g)\)-cages have a number of interesting properties. In this paper, we investigate various properties of cages, e.g., connectivity, the density of shortest cycles, bricks and braces.

Let \(G = (V, E)\) be a digraph with \(n\) vertices and \(m\) arcs without loops and multiarcs, \(V = \{v_1, v_2, \ldots, v_n\}\). Denote the outdegree and average \(2\)-outdegree of the vertex \(v_i\) by \(d^+_i\) and \(m^+_i\), respectively. Let \(A(G)\) be the adjacency matrix and \(D(G) = \text{diag}(d^+_1, d^+_2, \ldots, d^+_n)\) be the diagonal matrix with outdegrees of the vertices of the digraph \(G\). Then we call \(Q(G) = D(G) + A(G)\) the signless Laplacian matrix of \(G\). In this paper, we obtain some upper and lower bounds for the spectral radius of \(Q(G)\), which is called the signless Laplacian spectral radius of \(G\). We also show that some bounds involving outdegrees and the average \(2\)-outdegrees of the vertices of \(G\) can be obtained from our bounds.

Lee and Kong conjecture that if \(n \geq 1\) is an odd number, then \(St(a_1, a_0, \ldots, a_n)\) would be super edge-magic, and meanwhile they proved that the following graphs are super edge-magic: \(St(m,n)\) (\(n = 0 \mod (m+1)\)), \(St(1,k,n)\) (\(k = 1,2\) or \(n\)), \(St(2, k,n)\) (\(k = 2,3\)), \(St(1,1,k,n)\) (\(k = 2,3\)), \(St(k,2,2,n)\) (\(k = 1,2\)). In this paper, the conjecture is further discussed and it is proved that \(St(1,m,n)\), \(St(3,m,m+1)\), \(St(n,n+1,n+2)\) are super edge-magic, and under some conditions \(St(a_1, a_2, \ldots, a_{2n+1})\); \(St(a_1, a_2, \ldots, a_{4n+1})\), \(St(a_1, a_2, \ldots, a_{4n+3})\) are also super edge-magic.

We determine all connected odd graceful graphs of order \(\leq 6\). We show that if \(G\) is an odd graceful graph, then \(G \cup K_{m,n}\) is odd graceful for all \(m, n \geq 1\). We give an analogous statement to the graceful graphs statement, and we show that some families of graphs are odd graceful.

In this paper, we provide a method to obtain the lower bound on the number of distinct maximum genus embeddings of the complete bipartite graph \(K_{n,n}\) (\(n\) is an odd number), which, in some sense, improves the results of S. Stahl and H. Ren.

For positive integer \(n\), let \(f_3(n)\) be the least upper bound of the sums of the lengths of the sides of \(n\) cubes packed into a unit cube \(C\) in three dimensions in such a way that the smaller cubes have sides parallel to those of \(C\). In this paper, we improve the lower bound of \(f_3(n)\).

The transformation graph \(G^{+- -}\) of a graph \(G\) is the graph with vertex set \(V(G) \cup E(G)\), in which two vertices \(u\) and \(uv\) are joined by an edge if one of the following conditions holds: (i) \(u,v \in V(G)\) and they are adjacent in \(G\), (ii) \(u,v \in E(G)\) and they are not adjacent in \(G\), (iii) one of \(u\) and \(wv\) is in \(V(G)\) while the other is in \(E(G)\), and they are not incident in \(G\). In this paper, for any graph \(G\), we determine the independence number and the connectivity of \(G^{+- -}\). Furthermore, we show that for a graph \(G\) with no isolated vertices, \(G^{+- -}\) is hamiltonian if and only if \(G\) is not a star and \(G \not\in \{2K_2, K_2\}\).

We introduce quasi-almostmedian graphs as a natural nonbipartite generalization of almostmedian graphs. They are filling a gap between quasi-median graphs and quasi-semimedian graphs. We generalize some results of almostmedian graphs and deduce some results from a bigger class of quasi-semimedian graphs. The consequence of this is another characterization of almostmedian graphs as well as two new characterizations of quasi-median graphs.

In this note, we establish a convolution formula for Bernoulli polynomials in a new and brief way, and some known results are derived as a special case.

In this study, we define the generalized \(k\)-order Fibonacci matrix and the \(n \times n\) generalized Pascal matrix \(\mathcal{F}_n(GF)\) associated with generalized \(\mathcal{F}\)-nomial coefficients. We find the inverse of the generalized Pascal matrix \(\mathcal{F}_n(GF)\) associated with generalized \(\mathcal{F}\)-nomial coefficients. In the last section, we factorize this matrix via the generalized \(k\)-order Fibonacci matrix and give illustrative examples for these factorizations.

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let \(\mathcal{G}\) be the set of unicyclic graphs of order \(n\) with girth \(g\). For all integers \(n\) and \(g\) with \(5 \leq g \leq n – 6\), we determine the first \(|\frac{g}{2}| + 3\) spectral radii of unicyclic graphs in the set \(\mathcal{U}_n^g\).

In this paper, we consider labelings of graphs in which the label on an edge is the absolute value of the difference of its vertex labels. Such a labeling using \(\{0,1,2,\ldots,k-1\}\) is called \(k\)-equitable if the number of vertices (resp. edges) labeled \(i\) and the number of vertices (resp. edges) labeled \(j\) differ by at most one and is called \(k\)-balanced if the number of vertices labeled \(i\) and the number of edges labeled \(j\) differ by at most one. We determine which graphs in certain families are \(k\)-equitable or \(k\)-balanced and we give also some necessary conditions on these two labelings.

The study of chromatically unique graphs has been drawing much attention and many results are surveyed in \([4, 12, 13]\). The notion of adjoint polynomials of graphs was first introduced and applied to the study of the chromaticity of the complements of the graphs by Liu \([17]\) (see also \([4]\)). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu \([17]\) and Dong et al. \([4]\) respectively. In the paper, we shall utilize, among other things, these two invariants to investigate the chromaticity of the complement of the tadpole graphs \(C_n(P_m)\), the graph obtained from a path \(P_m\) and a cycle \(C_n\) by identifying a pendant vertex of the path with a vertex of the cycle. Let \(\bar{G}\) stand for the complement of a graph \(G\). We prove the following results:

1. The graph \(\overline{{{C}_{n-1}(P_2)}}\) is chromatically unique if and only if \(n \neq 5, 7\).
2. Almost every \(\overline{{C_n(P_m)}}\) is not chromatically unique, where \(n \geq 4\) and \(m \geq 2\).

An \(L(2,1)\)-labelling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x,y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x,y) = 2\), where \(d(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). The \((2,1)\)-labelling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labelling with \(\max\{f(v) : v \in V(G)\} = k\). Griggs and Yeh conjecture that \(\lambda(G) \leq \Delta^2\) for any simple graph with maximum degree \(\Delta \geq 2\). This article considers the graphs formed by the cartesian product of \(n\) (\(n \geq 2\) graphs. The new graph satisfies the above conjecture (with minor exceptions). Moreover, we generalize our results in [19].

In this study, we first define new sequences named \((s, t)\)-Jacobsthal and \((s, t)\) Jacobsthal-Lucas sequences. After that, by using these sequences, we establish \((s, t)\)-Jacobsthal and \((s, t)\) Jacobsthal-Lucas matrix sequences. Finally, we present some important relationships between these matrix sequences.

Several transformations about \(_\gamma F_6(1)\)-series are established by applying the modified Abel lemma on summation by parts. As a consequence, a reciprocal relation on balanced \(_3F_2(1)\)-series is derived, which may also be considered as a nonterminating extension of Saalschütz’s theorem (1891).

An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. A graph labeling of an almost-bipartite graph \(G\) with \(n\) edges that yields cyclic \(G\)-decompositions of the complete graph \(K_{2nt+1}\) was recently introduced by Blinco, El-Zanati, and Vanden Eynden. They called such a labeling a \(\gamma\)-labeling. Here we show that the class of almost-bipartite graphs obtained from a path with at least \(3\) edges by adding an edge joining distinct vertices of the path an even distance apart has a \(\gamma\)-labeling.

The locally twisted cube \(LTQ_n\) is an important variation of hypercube and possesses many desirable properties for interconnection networks. In this paper, we investigate the problem of embedding paths in faulty locally twisted cubes. We prove that a path of length \(l\) can be embedded between any two distinct vertices in \(LTQ_n – F\) for any faulty set \(F \subseteq V(LTQ_n) \cup E(LTQ_n)\) with \(|F| \leq n-3\) and any integer \(l\) with \(2^{n-1} \leq l \leq |V(LTQ_n – F)| – 1\) for any integer \(n > 3\). The result is tight with respect to the two bounds on path length \(l\) and faulty set size \(|F|\) for a successful embedding.

A \(G\)-design is a partition of \(E(K_v)\) in which each element induces a copy of \(G\). The existence of \(G\)-designs with the additional property that they contain no proper subsystems has been previously settled when \(G \in \{K_3, K_4 – e\}\). In this paper, the existence of \(P_m\)-designs which contain no proper subsystems is completely settled for every value of \(m\) and \(v\).

The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\). In this paper, we give a sharp lower bound on the Randić index of cacti with perfect matchings.

Let \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) be the \((2v+1)\)-dimensional affine-singular symplectic space over the finite field \(\mathbb{F}_q\) and let \(\text{ASp}_{2v+1}(\mathbb{F}_q)\) be the affine-singular symplectic group of degree \(2v+1\) over \(\mathcal{F}_q\). For any orbit \(O\) of flats under \(\text{ASp}_{2v+1}(\mathbb{F}_q)\), let \(\mathcal{L}\) be the set of all flats which are intersections of flats in \(O\) such that \(O \subseteq \mathcal{L}\) and assume the intersection of the empty set of flats in \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) is \(\mathbb{F}_q^{2v+1}\). By ordering \(\mathcal{L}\) by ordinary or reverse inclusion, two lattices are obtained. This article discusses the relations between different lattices, classifies their geometricity, and computes their characteristic polynomial.

Let \(\gamma_c(G)\) be the connected domination number of \(G\).A graph is \(k\)-\(\gamma_c\)-critical if \(\gamma_c(G) = k\) and \(\gamma_c(G + uv) < \gamma_c(G)\) for any nonadjacent pair of vertices \(u\) and \(v\) in the graph \(G\). In this paper, we show that the diameter of a \(k\)-\(\gamma_c\)-critical graph is at most \(k\) and this upper bound is sharp.

A \(b\)-coloring of a graph \(G\) by \(k\) colors is a proper \(k\)-coloring of the vertices of \(G\) such that in each color class there exists a vertex having neighbors in all the other \(k-1\) color classes. The \(b\)-chromatic number \(\varphi(G)\) of a graph \(G\) is the maximum \(k\) for which \(G\) has a \(b\)-coloring by \(k\) colors. This concept was introduced by R.W. Irving and D.F. Manlove in \(1999\). In this paper, we study the \(b\)-chromatic numbers of the cartesian products of paths and cycles with complete graphs and the cartesian product of two complete graphs.

Let \(K_{d,d}\) be a complete bipartite digraph. In this paper, we determine the exact value of the domination number in iterated line digraph of \(K_{d,d}\).

A total coloring of a simple graph \(G\) is called adjacent vertex distinguishing if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) differs from the set of colors assigned to the vertices and the edges incident to \(v\). In this paper, we shall prove that the adjacent vertex distinguishing total chromatic number of an outer plane graph with \(\Delta \leq 5\) is \(\Delta+2\) if \(G\) has two adjacent maximum degree vertices, otherwise it is \(\Delta+1\).

Let \(P_j(n)\) denote the number of representations of \(n\) as a sum of \(j\) pentagonal numbers. We obtain formulas for \(P_j(n)\) when \(j = 2\) and \(j = 3\).

Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. We study some variations of this concept in which the configuration of guards induce total dominating sets. We consider two models of the problem: one in which only one guard moves at a time and one in which all guards may move simultaneously. A number of upper and lower bounds are given for the number of guards required.

Let \(G\) be a finite graph and \(H\) be a subgraph of \(G\). If \(V(H) = V(G)\) then the subgraph is called a spanning subgraph of \(G\). A spanning subgraph \(H\) of \(G\) is called an \(F\)-factor if each component of \(H\) is isomorphic to \(F\). Further, if there exists a subgraph of \(G\) whose vertex set is \(V(G)\) and can be partitioned into \(F\)-factors, then it is called a \(\lambda\)-fold \(F\)-factor of \(G\), denoted by \(S_\lambda(1,F,G)\). A large set of \(\lambda\)-fold \(F\)-factors of \(G\), denoted by \(LS_\lambda(1,F,G)\), is a partition \(\{\mathcal{B}_i\}_i\) of all subgraphs of \(G\) isomorphic to \(F\), such that each \((X,\mathcal{B}_i)\) forms a \(\lambda\)-fold \(F\)-factor of \(G\). In this paper, we investigate \(LS_\lambda(1,K_{1,3},K_{v,v})\) for any index \(\lambda\) and obtain existence results for the cases \(v = 4t, 2t + 1, 12t+6\) and \(v \geq 3\).

In this paper, we give some interesting identities on the Bernoulli and the Euler numbers and polynomials by using reflection symmetric properties of Euler and Bernoulli polynomials. To derive our identities, we investigate some properties of the fermionic \(p\)-adic integrals on \(\mathbb{Z}_p\).

For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:

\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]

A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).

We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in [11] if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.

In \(1989\), Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\) and they obtained sufficient conditions for a graph to be hamiltonian with the implicit degrees. In this paper, we prove that if \(G\) is a \(2\)-connected graph of order \(n\) with \(\alpha(G) \leq n/2\) such that \(\text{id}(v) \geq (n-1)/2\) for each vertex \(v\) of \(G\), then \(G\) is hamiltonian with some exceptions.

The compact, Fredholm, and isometric weighted composition operators are characterized in this paper.

We discuss the chromaticity of one family of \(K_4\)-homeomorphs with exactly two non-adjacent paths of length two, where the other four paths are of length greater than or equal to three. We also give a sufficient and necessary condition for the graphs in the family to be chromatically unique.

In this paper, we deduced the following new Stirling series:

\[ n! \sim \sqrt{2n\pi} (\frac{n}{2})^n exp(\frac{1}{12n+1}[1 + \frac{1}{12n} (1+\frac{\frac{2}{5}}{n} + \frac{\frac{29}{150}}{n^2} – \frac{\frac{62}{2625}}{n^3} – \frac{\frac{9173}{157500}}{n^4} +\ldots )^{-1}]) ,\]

which is faster than the classical Stirling’s series.