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.