A family \(\mathcal{G}\) of connected graphs is said to be a family with constant metric dimension if \(\dim(G)\) does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we study the metric dimension of some plane graphs obtained from convex polytopes by attaching a pendant edge to each vertex of the outer cycle in a plane representation of these convex polytopes. We prove that the metric dimension of these plane graphs is constant and only three vertices, appropriately chosen, suffice to resolve all vertices of these classes of graphs. It is natural to ask for the characterization of graphs \(G\) that are plane representations of convex polytopes having the property that \(\dim(G) = \dim(G’)\), where \(G’\) is obtained from \(G\) by attaching a pendant edge to each vertex of the outer cycle of \(G\).

It is well known that the properties about the power sequences of different classes of sign pattern matrices may be very different. In this paper, we consider the base of primitive nonpowerful zero-symmetric square sign pattern matrices without nonzero diagonal entry. The base set is shown to be \(\{2, 3, \ldots, 2n – 1\}\); the extremal sign pattern matrices with base \(2n – 1\) are characterized. As well, for the sign patterns with order \(3\), the sign patterns with bases \(3\), \(4\), \(5\) are characterized, respectively.

In this note, we study clique number, chromatic number,domination number and independence number of the intersection graph of subspaces of a finite dimensional vector space over a finite field.

A vertex-colored graph \(G\) is rainbow connected, if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of a connected graph \(G\), denoted \(\mathrm{rvc}(G)\), is the smallest number of colors that are needed in order to make \(G\) rainbow vertex connected. In this paper, we show that \(\mathrm{rvc}(G) \leq k\), if \(|E(G)| \geq \binom{n-k}{2} + k\), for \(k = 2, 3, n-4, n-5, n-6\). These bounds are sharp.

In order to find more sufficient conditions for the existence of hamiltonian cycles of graphs, Zhu, Li, and Deng proposed the definition of implicit degree of a vertex. In this paper, we consider the relationship between implicit degrees of vertices and the hamiltonicity of graphs, and obtain that: If the implicit degree sum for each pair of nonadjacent vertices of an induced claw or an induced modified claw in a \(2\)-connected graph \(G\) is more than or equal to \(|V(G)| – 1\), then \(G\) is hamiltonian with some exceptions. This extends a previous result of Cai et al. [J. Cai, H. Li and W. Ning, An implicit degree condition for hamiltonian cycles, Ars Combin. \(108 (2013) 365-378.]\) on the existence of hamiltonian cycles.

The general vertex-distinguishing total chromatic number of a graph \(G\) is the minimum integer \(k\), for which the vertices and edges of \(G\) are colored using \(k\) colors such that there are no two vertices possessing the same color-set, where a color-set of a vertex is a set of colors of the vertex and its incident edges. In this paper, we discuss the general vertex-distinguishing total chromatic number of complete bipartite graphs \(K_{m,n}\), and obtain the exact value of this number for some cases in terms of \(m\) and \(n\). Particularly, we give the bounds of this number for \(K_{n,n}\).

In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number.

Let \(G = (V, E)\) be a graph and \(\phi: V \cup E \to \{1, 2, \ldots, \alpha\}\) be a proper \(\alpha\)-total coloring of \(G\). Let \(f(v)\) denote the sum of the color on vertex \(v\) and the colors on the edges incident with \(v\). A neighbor sum distinguishing \(\alpha\)-total coloring of \(G\) is a proper \(\alpha\)-total coloring of \(G\) such that for each edge \(uv \in E(G)\), \(f(u) \neq f(v)\). Pileeniak and Woźniak first introduced this coloring and conjectured that such coloring exists for any simple graph \(G\) with maximum degree \(\Delta(G)\) if \(\alpha \geq \Delta(G) + 3\). The maximum average degree of \(G\) is the maximum of the average degree of its non-empty subgraphs, which is denoted by \(mad(G)\). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that this conjecture holds for graphs with larger maximum average degree in their list versions. More precisely, we prove that if \(G\) is a graph with \(\Delta(G) \geq 11\) and \(mad(G) < 5\), then \(ch''_{\Sigma}(G) \leq \Delta(G) + 3\), where \(ch''_{\Sigma}(G)\) is the neighbor sum distinguishing total choosability of \(G\).

Let \(\mathcal{K}\) be a family of sets in \(\mathbb{R}^d\) and let \(k\) be a fixed natural number. Assume that every countable subfamily of \(K\) has an intersection expressible as a union of \(k\) starshaped sets, each having a \(d\)-dimensional kernel. Then \(S = \cap \{K : K \in \mathcal{K}\}\) is nonempty and is expressible as a union of \(k\) such starshaped sets.

If members of \(K\) are compact and every finite subfamily of \(\mathcal{K}\) has as its intersection a union of \(k\) starshaped sets, then \(S\) again is a union of \(k\) starshaped sets. An analogous result holds for unions of \(k\) convex sets. Finally, dual results hold for unions of subfamilies of \(\mathcal{K}\).

We give relationships among the binomial coefficients, the Bemoulli numbers and the Stirling numbers, These relations are derived from the translation formulae in the linear discrete systems in Shin-Naito \([8]\).

In this paper, we give the continued fraction expansions of the ordinary generating functions of the derangement polynomials of types \(A\) and \(B\) in a unified manner. Our proof is based on their exponential generating functions and the theory of exponential Riordan arrays.

A graph is called End-regular if its endomorphism monoid is regular. Which graphs are End-regular? This is an open question and difficult to obtain a general answer. In the present paper, we investigate the End-regularity of graphs obtained by adding or deleting vertices from End-regular graphs. As an application, we show that the non-commuting graphs of \(AC\)-groups are End-regular.

In this paper, we study some identities of Barnes-type Genocchi polynomials. We derive those identities by using the fermionic \(p\)-adic integral on \(\mathbb{Z}_p\).
In \([13]\), D.S. Kim and T. Kim established some identities of higher-order Bernoulli and Euler polynomials arising from Bernoulli and Euler basis, respectively. Using the idea developed in \([13]\), we study various identities of special polynomials arising from Barnes-type Genocchi basis.

Suppose that the vertex set of a graph \(G\) is \(V(G) = \{v_1, \ldots, v_n\}\). Then we denote by \({Tr_G}(v_i)\) the sum of distances between \(v_i\) and all other vertices of \(G\). Let \({Tr}(G)\) be the \(n \times n\) diagonal matrix with its \((i,i)\)-entry equal to \({Tr_G}(v_i)\) and \(D(G)\) be the distance matrix of \(G\). Then \(L_p(G) = {Tr}(G) – D(G)\) is the distance Laplacian matrix of \(G\). The largest eigenvalues of \(D(G)\) and \(L_p(G)\) are called distance spectral and distance Laplacian spectral radius of \(G\), respectively. In this paper, we describe the unique graph with maximum distance and distance Laplacian spectral radius among all connected graphs of order \(n\) with given cut edges.

A radio labeling of a connected graph \(G\) of diameter \(d\) is a mapping \(f: V(G) \to \{0, 1, 2, \ldots\}\) such that \(d(u, v) + |f(u) – f(v)| \geq d + 1\) for each pair of distinct vertices \(u\) and \(v\) of \(G\), where \(d(u, v)\) is the distance between \(u\) and \(v\). The value \(rn(f)\) of a radio labeling \(f\) is the maximum label assigned by \(f\) to a vertex of \(G\). The radio number \(rn(G)\) of \(G\) is the minimum value of \(rn(f)\) taken over all radio labelings \(f\) of \(G\). A caterpillar \(C_{m,t}\) is a special tree that consists of a path \(x_1x_2 \ldots x_m\) (\(m \geq 3\)), with some pendant vertices adjacent to the inner vertices \(x_2, x_3, \ldots, x_{m-1}\). If \(d(x_i) = t\) (the degree of \(x_i\)) for \(i = 2, 3, \ldots, m-1\), then the caterpillar is called standard. In this paper, we determine the exact value of the radio number of \(C_{m,t}\) for all integers \(m \geq 4\) and \(t \geq 2\), and explicitly construct an optimal radio labeling. Our results show that the radio number and the construction of optimal radio labeling of paths are special cases of \(C_{m,t}\) with \(t = 2\).

Graph theory, with its diverse applications in theoretical computer science and in natural sciences (chemistry, biology), is becoming an important component of mathematics. Recently, the concepts of new Zagreb eccentricity indices were introduced. These indices were defined for any graph \(G\), as follows: \(M_1^*(G) = \sum_{e_{uv} \in E(G)} [\varepsilon_G(u) + \varepsilon_G(v)]\), \(M_1^{**}(G) = \sum_{v \in V(G)} [\varepsilon_G(v)]^2\), and \(M_2^*(G) = \sum_{e_{uv} \in E(G)} |\varepsilon_G(u) – \varepsilon_G(v)|\), where \(\varepsilon_G(u)\) is the eccentricity value of vertex \(u\) in the graph \(G\). In this paper, new Zagreb eccentricity indices \(M_1^*(G)\), \(M_1^{**}(G)\), and \(M_2^*(G)\) of cycles related graphs, namely gear, friendship, and corona graphs, are determined. Then, a programming code finding values of new Zagreb indices of any graph is offered.

Bizley [J. Inst. Actuar. 80 (1954), 55-62] studied a generalization of Dyck paths from \((0,0)\) to \((pn, gn)\) (\(\gcd(p,q) = 1\)), which never go below the line \(py = qx\) and are made of steps in \(\{(0, 1), (1,0)\}\), called the step set, and calculated the number of such paths. In this paper, we mainly generalize Bizley’s results to an arbitrary step set \(S\). We call these paths \(S\)-\((p,q)\)-Dyck paths, and give explicit enumeration formulas for such paths. In addition, we provide a proof of these formulas using the method presented in Gessel [J. Combin. Theory Ser. A 28 (1980), no. 3, 321-337]. As applications, we calculate some examples which generalize the classical Schröder and Motzkin numbers.

A \(2\)-rainbow dominating function of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V(G)\) with \(f(v) = \emptyset\) the condition \(\bigcup_{u \in N(v)} f(u) = \{1,2\}\) is fulfilled, where \(N(v)\) is the open neighborhood of \(v\). A rainbow dominating function \(f\) is said to be a rainbow restrained domination function if the induced subgraph of \(G\) by the vertices with label \(\emptyset\) has no isolated vertex. The weight of a rainbow restrained dominating function is the value \(w(f) = \sum_{v \in V(G)} |f(v)|\). The minimum weight of a rainbow restrained dominating function of \(G\) is called the rainbow restrained domination number of \(G\). In this paper, we continue the study of the rainbow restrained domination number. First, we classify all graphs \(G\) of order \(n\) whose rainbow restrained domination number is \(n-1\). Then, we establish an upper bound on the rainbow restrained domination number of trees.

The entire chromatic number \(\chi_c(G)\) of a plane graph \(G(V, E, F)\) is the minimum number of colors such that any two distinct adjacent or incident elements receive different colors in \(V(G) \cup E(G) \cup F(G)\). A plane graph \(G\) is called a \(1\)-tree if there exists a vertex \(u \in V(G)\) such that \(G – u\) is a forest. In this paper, it is proved that if \(G\) is a \(2\)-connected \(1\)-tree with \(\Delta(G) \geq 6\), then the entire chromatic number of \(G\) is \(\Delta(G) + 1\), where \(\Delta(G)\) is the maximum degree of \(G\).

In this paper, we compute various finite sums that alternate according to \((-1)^{\binom{n}{k}}\) involving the generalized Fibonacci and Lucas numbers for \(k = 3, 4, 5\) and even \(k\) of the form \(2^m\) with \(m \geq 1\).

In this paper, we introduce a special \((k_1A_1, k_2A_2, k_3A_3)\)-edge colouring of a graph. We shall show that for special graphs and special values of \(k_i\), \(i = 1, 2, 3\), the number of such colourings generalizes the well-known Pell numbers. Using this graph interpretation, we give a direct formula for the generalized Pell numbers. Moreover, we show some identities for these numbers.

The multiplicatively weighted Harary index (\(Hy\)-index) is a new distance-based graph invariant, which was introduced and studied by Deng et al. in [1]. For a connected graph \(G\), the multiplicatively weighted Harary index of \(G\) is defined as \(H_M(G) = \sum\limits_{\{u,v\} \subseteq V(G)} \frac{d_G(u) \cdot d_G(v)}{d_G(u,v)}\), where \(d_G(x)\) denotes the degree of vertex \(x\) and \(d_G(s,t)\) denotes the distance between vertices \(s\) and \(t\) in \(G\). In this paper, we first study a new vertex degree-based graph invariant \(M_2 – \frac{1}{2}M_1\), where \(M_1\) and \(M_2\) are ordinary Zagreb indices. We characterize the trees attaining maximum value of \(M_2 – 4M_1\) among all trees of given order. As applications, we obtain a new proof of Deng et al.’s results on trees with extremal \(H_M\)-index among all trees of given order.

In the current work, the author presents a symbolic algorithm for finding the determinant of any general nonsingular cyclic heptadiagonal matrices and the inverse of anti-cyclic heptadiagonal matrices. The algorithms are mainly based on the work presented in [A. A. Karawia, A New algorithm for inverting general cyclic heptadiagonal matrices recursively, arXiv:1011.2306v1, ICS/SCII]. The symbolic algorithms are suited for implementation using Computer Algebra Systems (CAS) such as MATLAB, MAPLE, and MATHEMATICA. An illustrative example is given.

The Wiener index of a graph is a distance-based topological index defined as the sum of distances between all pairs of vertices. In this paper, two explicit expressions for the expected value of the Wiener indices of two types of random polygonal chains are obtained.

In \([8]\), the author introduced the notion of burst errors for \(2\)-dimensional array coding systems. Also, in \([10]\), the author introduced a series of metrics called Lee-RT-Jain-Metric (LRTJ\)-metric) \([3]\) for array codes, which is a generalization of both classical Lee metric \([12]\) and array \(RT\) metric \([14]\). In this paper, we obtain sufficient conditions on the parameters of array codes equipped with \(LRTJ\)-metric for the identification and correction of burst array errors.

The concept of exterior degree of a finite group \(G\) is introduced by the author in a joint paper [13], which is the probability of randomly selecting two elements \(g\) and \(h\) in \(G\) such that \(g\wedge h = 1\). In the present paper, a necessary and sufficient condition is given for a non-cyclic group when its exterior degree achieves the upper bound \((p^2 + p – 1)/p^3\), where \(p\) is the smallest prime number dividing the order of \(G\). We also compute the exterior degree of all extra-special \(p\)-groups. Finally, for an extra-special \(p\)-group \(H\) and a group \(G\) where \(G/Z^\wedge(G)\) is a \(p\)-group, we will show that \(d^\wedge(G) = d^\wedge(H)\) if and only if \(G/Z^\wedge(G) \cong H/Z^\wedge(H)\), provided that \(d^\wedge(G) \neq 11/32\).

Let \(G\) be a unicyclic graph on \(n \geq 3\) vertices. Let \(A(G)\) be the adjacency matrix of \(G\). The eigenvalues of \(A(G)\) are denoted by \(\lambda_1(G) \geq \lambda_2(G) \geq \cdots \geq \lambda_n(G)\), which are called the eigenvalues of \(G\). Let the unicyclic graphs \(G\) on \(n\) vertices be ordered by their least eigenvalues \(\lambda_n(G)\) in non-decreasing order. For \(n \geq 14\), the first six graphs in this order are determined.

Hyperdomination in hypergraphs was defined by J. John Arul Singh and R. Kala in [3]. Let \(X = \{a_1, a_2, \ldots, a_n\}\) be a finite set and let \(\mathcal{E} = \{E_1, E_2, \ldots, E_m\}\) be a family of subsets of \(X\). \(H = (X, \mathcal{E})\)is said to be a hypergraph if (1) \(E_i \neq \phi\), \(1 \leq i \leq m\), and (2) \(\bigcup_{i=1}^{m} E_i = X\). The elements \(x_1, x_2, \ldots, x_n\) are called the vertices and the sets \(E_1, E_2, \ldots, E_m\) are called the edges. A set \(D \subset X\) is called a hyperdominating set if for each \(v \in X – D\) there exist some edge \(E\) containing \(v\) with \(|E| \geq 2\) such that \(E – v \subset D \neq D\). The hyperdomination number is the minimum cardinality of all hyperdominating sets. In this paper, a finite group is viewed as a hypergraph with vertex set as the elements of the group and edge set as the set of all subgroups of the group. We obtain several bounds for hyperdomination number of finite groups and characterise the extremal graphs in some cases.

Let \(G\) be a simple graph with edge ideal \(I(G)\). In this article, we study the number of pairwise \(3\)-disjoint edges of cycles and complements of triangle-free graphs. Using that, we determine the Castelnuovo-Mumford regularity of \(R/I(G)\) for the above classes of graphs according to the number of pairwise \(3\)-disjoint edges.

The Merrifield-Simmons index \(i(G)\) of a graph \(G\) is defined as the total number of independent sets of \(G\). A connected graph \(G = (V,E)\) is called a quasi-unicyclic graph if there exists a vertex \(u_0 \in V\) such that \(G – u_0\) is a unicyclic graph. Denote by \(\mathcal{U}(n,d_0)\) the set of quasi-unicyclic graphs of order \(n\) with \(G – u_0\) being a unicyclic graph and \(d_G(u_0) = d_0\). In this paper, we characterize the quasi-unicyclic graphs with the smallest, the second-smallest, the largest, and the second-largest Merrifield-Simmons indices, respectively, in \(\mathcal{U}(n, d_0)\).

A unicyclic map is a rooted planar map such that there is only one cycle which is the boundary of the unique inner face (the inner face contains no trees) and the root-vertex is on the cycle. In this paper we investigate the number of unicyclic maps and present some formulae for such maps with up to three parameters: the number of edges and the valencies of the root-vertex and the root-face.

Brualdi and Massey in \(1993\) posed two conjectures regarding the upper bound for incidence coloring number of graphs in terms of maximum degree. In this paper among some results, we prove these conjectures for some classes of graphs with maximum degree \(4\).

P. Erdős, F. Harary, and M. Klawe studied the \(K_n\)-residual graph and came up with some conjectures and conclusions about the \(m-K_n\)-residual graph. For connected \(m-K_2\)-residual graphs, they constructed an \(m-K_2\)-residual graph of order \(3m+2\) and proposed that \(3m+2\) is the minimum order, which remained unproven. In this paper, using operation properties of sets and other methods, we prove that the minimum order of connected \(m-K_2\)-residual graphs is indeed \(3m+2\).

In this paper, we present explicit formulas for domination numbers of equidistant \(m\)-cactus chains and find the corresponding minimum dominating sets. For an arbitrary \(m\)-cactus chain, we establish the lower and upper bounds for its domination number. We find some extremal chains with respect to this graph invariant.

A strongly connected digraph \(D\) is said to be maximally arc connected if its arc-connectivity \(\lambda(D)\) attains its minimum degree \(\delta(D)\). For any vertex \(x\) of \(D\), the set \(\{x^g \mid g \in \text{Aut}(D)\}\) is called an orbit of \(\text{Aut}(D)\). Liu and Meng [ Fengxia Liu, Jixiang Meng, Edge-Connectivity of regular graphs with two orbits, Discrete Math. \(308 (2008) 3711-3717 \)] proved that the edge-connectivity of a \(k\)-regular connected graph with two orbits and girth \(\geq 5\) attains its regular degree \(k\). In the present paper, we prove the existence of \(k\)-regular \(m\)-arc-connected digraphs with two orbits for some given integer \(k\) and \(m\). Furthermore, we prove that the \(k\)-regular connected digraphs with two orbits, satisfying girth \( \geq k\) are maximally arc connected. Finally, we give an example to show that the girth bound \(k\) is best possible.

Let \(G\) be a graph with a vertex coloring. A colorful path is a path with \(\chi(G)\) vertices, in which the vertices have different colors. A colorful path starting at vertex \(v\) is a colorful \(v\)-path. We show that for every graph \(G\) and given vertex \(v\) of \(G\), there exists a proper vertex coloring of \(G\) with a colorful path starting at \(v\). Let \(G\) be a connected graph with maximum degree \(\Delta(G)\) and \(|V(G)| \geq 2\). We prove that there exists a proper \((\chi(G) + \Delta(G) – 1)\)-coloring of \(G\) such that for every \(v \in V(G)\), there is a colorful \(v\)-path.

Let \(\mathcal{B}(n,d)\) be the set of bicyclic graphs with both \(n\) vertices and diameter \(d\), and let \(\theta^*\) consist of three paths \(u_0w_1v_0\), \(u_0w_2v_0\), and \(u_0w_3v_0\). For four nonnegative integers \(n,d,k,j\) satisfying \(n \geq d+3\), \(d=k+j+2\), we let \(B(n,d;k,j)\) denote the bicyclic graph obtained from \(\theta^*\) by attaching a path of length \(k\) to \(u_0\), attaching a path of length \(j\) to vertex \(v_0\) and \(n-d-3\) pendant edges to \(w_0\), and let \(\mathcal{B}(n,d;k,j) = \{B(n,d;k,j) \mid k+j \geq 1\}\). In this paper, the extremal graphs with the minimal least eigenvalue among all graphs in \(\mathcal{B}(n,d;k,j)\) are well characterized, and some structural characterizations about the extremal graphs with the minimal least eigenvalue among all graphs in \(\mathcal{B}(n,d)\) are presented as well.

If \(G = (V, E)\) is a simple connected graph and \(a, b \in V\), then a shortest \((a – b)\) path is called an \((a – b)\)-geodesic. A set \(X \subseteq V\) is called weakly convex in \(G\) if for every two vertices \(a, b \in X\) there exists an \((a – b)\)-geodesic whose all vertices belong to \(X\). A set \(X\) is convex in \(G\) if for every \(a, b \in X\) all vertices from every \((a – b)\)-geodesic belong to \(X\). The weakly convex domination number of a graph \(G\) is the minimum cardinality of a weakly convex dominating set in \(G\), while the convex domination number of a graph \(G\) is the minimum cardinality of a convex dominating set in \(G\). In this paper, we consider weakly convex and convex domination numbers of Cartesian products, joins, and coronas of some classes of graphs.

Using a new way to label edges in a bicoloured ordered tree,we introduce a bijection between bicoloured ordered trees and non-nesting partitions. Consequently, enumerative results of non-nesting partitions are derived. Together with another bijection given before, we obtain a bijection between non-nesting partitions and non-crossing partitions specified with four parameters.

Lee and Wei defined super vertex-graceful labeling in 2006. In this paper, the generalized Butterfly Graph \(B_{n}^t\) and \(C_n^{(t)}\) graph are discussed. The generalized butterfly Graph \(B_{n}^t\) is super vertex-graceful when \(t\) (\(t > 0\)) is even, \(B_{n}^0\) is super vertex-graceful when \(n \equiv 0, 3 \pmod{4}\); For \(C_3^{(t)}\), there are: \(C_3^{(t)}\) is super vertex-graceful if and only if \(t = 1, 2, 3, 5, 7\). Moreover, we propose two conjectures on super vertex-graceful labeling.

An adjacent vertex distinguishing edge coloring, or an avd-coloring, of a simple graph \(G\) is a proper edge coloring of \(G\) such that for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the edges incident to \(u\) differs from the set of colors assigned to the edges incident to \(v\). In this paper, we prove that graphs with maximum degree \(3\) and with no isolated edges partly satisfy the adjacent vertex distinguishing edge coloring conjecture.

An \(L(2,1)\)-labeling 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 vertices \(x\) and \(y\) in \(G\). The \(L(2,1)\)-labeling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labeling with \(\max\{f(v) : v \in V(G)\} = k\). We consider Cartesian sums of graphs and derive, both, lower and upper bounds for the \(L(2,1)\)-labeling number of this class of graphs; we use two approaches to derive the upper bounds for the \(L(2,1)\)-labeling number and both approaches improve previously known upper bounds. We also present several approximation algorithms for computing \(L(2,1)\)-labelings for Cartesian sum graphs.

Assume that \(G = (V, E)\) is an undirected graph with vertex set \(V\) and edge set \(E\). The ball \(B_r(v)\) denotes the vertices within graphical distance \(r\) from \(v\). A subset \(C \subseteq V\) is called an \(r\)-locating-dominating code if the sets \(I_r(v) = B_r(v) \cap C\) are distinct and non-empty for all \(v \in V \setminus C\). A code \(C\) is an \(r\)-identifying code if the sets \(I_r(v) = B_r(v) \cap C\) are distinct and non-empty for all vertices \(v \in V\). We study \(r\)-locating-dominating codes in the infinite king grid and, in particular, show that there is an \(r\)-locating-dominating code such that every \(r\)-identifying code has larger density. The infinite king grid is the graph with vertex set \(\mathbb{Z}^2\) and edge set \(\{(x_1, y_1), (x_2, y_2) \mid |x_1 – x_2| \leq 1, |y_1 – y_2| \leq 1, (x_1, y_1) \neq (x_2, y_2)\}\).

An antimagic labeling of a graph with \(n\) vertices and \(m\) edges is a bijection from the set of edges to the integers \(1, 2, \ldots, m\) such that all \(n\) vertex sums are pairwise distinct. For a cycle \(C_n\) of length \(n\), the \(k\)-th power of \(C_n\), denoted by \(C_n^k\), is the supergraph formed by adding an edge between all pairs of vertices of \(C_n\) with distance at most \(k\). Antimagic labelings for \(C_n^k\) are given where \(k = 2, 3, 4\).

In 1990, Kostochka and Sidorenko proposed studying the smallest number of list-colorings of a graph \(G\) among all assignments of lists of a given size \(n\) to its vertices. We say a graph \(G\) is \(n\)-monophilic if this number is minimized when identical \(n\)-color lists are assigned to all vertices of \(G\). Kostochka and Sidorenko observed that all chordal graphs are \(n\)-monophilic for all \(n\). Donner (1992) showed that every graph is \(n\)-monophilic for all sufficiently large \(n\). We prove that all cycles are \(n\)-monophilic for all \(n\); we give a complete characterization of \(2\)-monophilic graphs (which turns out to be similar to the characterization of \(2\)-choosable graphs given by Erdős, Rubin, and Taylor in 1980); and for every \(n\) we construct a graph that is \(n\)-choosable but not \(n\)-monophilic.

An adjacent vertex distinguishing total coloring of a graph \(G\) is a proper total coloring of \(G\) such that no two adjacent vertices are incident to the same set of colors. The minimum number of colors needed for such a coloring is denoted by \(\chi_{at}(G)\). In this note, we prove that \(\chi_{at}(G) = 5\) for some cubic graphs.

In this paper, a generalized notion of the fixed point property,namely the \(n\)-fixed point property, for posets is discussed. The \(n\)-fixed point property is proved to be equivalent to the fixed point property in lattices. Further, it is shown that a poset of finite width has the \(n\)-fixed point property for some natural number \(n\) if and only if every maximal chain in it is a complete lattice.

A graph is said to be equitably \(k\)-colorable if the vertex set \(V(G)\) can be partitioned into \(k\) independent subsets \(V_1, V_2, \ldots, V_k\) such that \(||V_i| – |V_j|| \leq 1\) (\(1 \leq i, j \leq k\)). A graph \(G\) is equitably \(k\)-choosable if, for any given \(k\)-uniform list assignment \(L\), \(G\) is \(L\)-colorable and each color appears on at most \(\lceil \frac{|V(G)|}{k} \rceil\) vertices. In this paper, we prove that if \(G\) is a graph such that \(\mathrm{mad}(G) \leq 3\), then \(G\) is equitably \(k\)-colorable and equitably \(k\)-choosable where \(k \geq \max\{\Delta(G), 5\}\).

For a vertex \(v\) of a graph \(G\), the unlabeled subgraph \(G-v\) is called a card of \(G\). We prove that the number of isolated vertices and the number of components of an \(n\)-vertex graph \(G\) can be determined from any collection of \(n-2\) of its cards for \(n \geq 10\). It is also proved that if two graphs of order \(n \geq 6\) have \(n-2\) cards in common, then the number of edges in them differs by at most one.

The Hosoya index \(z(G)\) of a graph \(G\) is defined as the total number of matchings of \(G\), and the Merrifield-Simmons index \(i(G)\) of a graph \(G\) is defined as the total number of independent sets of \(G\). Although there are many known results on these two indices, there exist few on a given class of graphs with perfect matchings. In this paper, we first introduce two new strengthened transformations. Then we characterize the extremal unicyclic graphs with perfect matching which have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield-Simmons index, respectively.

The aim of this paper is to introduce the notions of \(f\)-derivation and symmetric bi-derivation in \(c\)-subtraction algebras and to study some properties of these derivations.

A book-embedding of a graph \(G\) consists of placing the vertices of \(G\) on a spine and assigning edges of the graph to pages so that edges assigned to the same page without crossing. In this paper,we propose schemes to embed the connected triple-loop networks with even cardinality in books, then we give upper bounds of page number of some multi-loop networks.

Determining the size of a maximum independent set of a graph \(G\), denoted by \(\alpha(G)\), is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of \(\alpha(G)\) for special classes of graphs. This paper is aimed towards studying this problem for the class of generalized Petersen graphs. We find new upper and lower bounds and some exact values for \(\alpha(P(n,k))\). With a computer program, we have obtained exact values for each \(n 2k\). We prove this conjecture for some cases. In particular, we show that if \(n > 3k\), the conjecture is valid. We checked the conjecture with our table for \(n < 78\) and found no inconsistency. Finally, we show that for every fixed \(k\), \(\alpha(P(n,k))\) can be computed using an algorithm with running time \(O(n)\).

In this paper we give the solutions of finding maximum packings and minimum coverings of \(\lambda\)-fold complete symmetric digraphs with \(6\)-circuits.

In recent researches on a discriminant for polynomials, I faced a recursive (combinatorial) sequence \(\lambda_{n,m}\) whose first four terms and identities are \(\lambda_{0,m} := \binom{m}{0}\), \(\lambda_{1,m} := \binom{m}{1}=\binom{m}{m-1}\), \(\lambda_{2,m} := {\binom{m}{2}}^2 – \binom{m}{2}=\binom{m+1}{m-1}\), and \(\lambda_{3,m} = {\binom{m}{1}}^3 – 2\binom{m}{1}\binom{m}{2} + \binom{m}{3}=\binom{m+2}{m-1}\). In this paper, I introduce this sequence, prove an identity concerning it, and leave a problem and a conjecture regarding its properties.

Let \(\mu_1, \mu_2, \ldots, \mu_n\) be the eigenvalues of the sum-connectivity matrix of a graph \(G\). The sum-connectivity spectral radius of \(G\) is the largest eigenvalue of its sum-connectivity matrix, and the sum-connectivity Estrada index of \(G\) is defined as \(\mathrm{SEE}(G) = \sum_{i=1}^{n} e^{\mu_i}\). In this paper, we obtain some results about the sum-connectivity spectral radius of graphs. In addition, we give some upper and lower bounds on the sum-connectivity Estrada index of graph \(G\), as well as some relations between \(\mathrm{SEE}\) and sum-connectivity energy. Moreover, we characterize that the star has maximum sum-connectivity Estrada index among trees on \(n\) vertices.

Let \(G(n;\theta_{2k+1})\) denote the class of non-bipartite graphs on \(n\) vertices containing no \(\theta_{2k+1}\)-graph and let \(f(n; \theta_{2k+1}) = \max\{\varepsilon(G) : G \in \mathcal{G}(n;\theta_{2k+1})\}\). In this paper, we determine \(f(n; 0_5)\), by proving that for \(n \geq 11\), \(f(n; 0_5) \leq \lfloor\frac{(n-1)^2}{4}\rfloor + 1\). Further, the bound is best possible. Our result confirms the validity of the conjecture made in [1], “Some extremal problems in graph theory”, Ph.D. thesis, Curtin University of Technology, Australia (2007).

Let \(G\) be a cactus, where all blocks of \(G\) are either edges or cycles. Denote \(\mathcal{G}(n,r)\) the set of cactuses of order \(n\) and with \(r\) cycles. In this paper, we present a unified approach to the extremal cactuses for the Schultz and the modified Schultz indices.

In this paper, I study the Eulerian numbers \((A(m,k))_{k=1}^{m}\) and prove the relationship between \(\sum_{i=1}^{n}{i^m}\) and \((A(m,k))_{k=1}^{m}\), to be \(\sum_{i=1}^{n}{i^m} = \sum_{k=1}^m A(m,k)\binom{m+k}{m+1}\).

An \((s, t)\)-spread in a finite vector space \(V = V(n, q)\) is a collection \(\mathcal{F}\) of \(t\)-dimensional subspaces of \(V\) with the property that every \(s\)-dimensional subspace of \(V\) is contained in exactly one member of \(F\). It is remarkable that no \((s, t)\)-spreads have been found yet, except in the case \(s = 1\).
In this note, the concept of an \(\alpha\)-point to a \((2,3)\)-spread \(\mathcal{F}\) in \(V = V(7,2)\) is introduced. A classical result of Thomas, applied to the vector space \(V\), states that all points of \(V\) cannot be \(\alpha\)-points to a given \((2,3)\)-spread \(\mathcal{F}\) in \(V\). In this note, we strengthen this result by proving that every \(6\)-dimensional subspace of \(V\) must contain at least one point that is not an \(\alpha\)-point to a given \((2, 3)\)-spread.

We construct explicitly the automorphism group of the folded hypercube \(FQ_n\) of dimension \(n > 3\), as a semidirect product of \(N\) by \(M\), where \(N\) is isomorphic to the Abelian group \(\mathbb{Z}_2^{n}\), and \(M\) is isomorphic to \(\mathrm{Sym}(n+1)\), the symmetric group of degree \(n+1\). Then, we will show that the folded hypercube \(FQ_n\) is a symmetric graph.

The Merrifield-Simmons index \(\sigma(G)\) of a graph \(G\) is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent vertex sets of \(G\). A tree is called an \(r\)-leaf tree if it contains \(r\) vertices with degree one. In this paper, we obtain the smallest Merrifield-Simmons index among all trees with \(n\) vertices and exactly six leaves, and characterize the corresponding extremal graph.

A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\dim(G)\) is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). The metric dimension of some classes of plane graphs has been determined in \([2], [3],[ 4], [9], [10], [14], [22]\). In this paper, we extend this study by considering some classes of plane graphs which are rotationally-symmetric. It is natural to ask for the characterization of classes of rotationally-symmetric plane graphs with constant metric dimension.

is almost locally connected if \(B(G)\) is an independent set and for any \(x \in B(G)\), there is a vertex \(y\) in \(V(G) \setminus \{x\}\) such that \(N(x) \cup \{y\}\) induces a connected subgraph of \(G\), where \(B(G)\) denotes the set of vertices of \(G\) that are not locally connected. In this paper, we prove that an almost locally connected claw-free graph on at least \(4\) vertices is Hamilton-connected if and only if it is \(3\)-connected. This generalizes a result by Asratian that a locally connected claw-free graph on at least \(4\) vertices is Hamilton-connected if and only if it is \(3\)-connected [Journal of Graph Theory \(23 (1996) 191-201\)].

We give new expressions for Stirling numbers, and some partial sums of powers and products.

A star coloring of an undirected graph \(G\) is a proper vertex coloring of \(G\) such that any path on four vertices in \(G\) is not bicolored. The star chromatic number \(\chi_s(G)\) of an undirected graph \(G\) is the smallest integer \(k\) for which \(G\) admits a star coloring with \(k\) colors. In this paper, the star chromatic numbers for some infinite subgraphs of Cartesian products of paths and cycles are established. In particular, we show that \(\chi_s(P_i \Box C_j) = 5\) for \(i, j \geq 4\) and \(\chi_s(C_i \Box C_j) = 5\) for \(i, j \geq 30\). We also show that \(\chi_s(P_i \Box P_j \Box P_k) = 6\) for \(i, j, k \geq 4\), \(\chi_s(C_{3} \Box C_{3} \Box C_k) = 7\) for \(k \geq 3\), and \(\chi_s(C_{4i} \Box C_{4j} \Box P_{4k} \Box C_{4l}) \leq 9\) for \(i, j, k, l \geq 1\). Furthermore, we give the star chromatic numbers of \(d\)-dimensional hypercubes for \(d \leq 6\).

Mixed connectivity is a generalization of vertex and edge connectivity. A graph is \((p,0)\)-connected, \(p \geq 0\), if the graph remains connected after removal of any \(p – 1\) vertices. A graph is \((p,q)\)-connected, \(p \geq 0\), \(q \geq 0\), if it remains connected after removal of any \(p\) vertices and any \(q – 1\) edges. Cartesian graph bundles are graphs that generalize both covering graphs and Cartesian graph products. It is shown that if graph \(F\) is \((p_F, q_F)\)-connected and graph \(B\) is \((p_B, q_B)\)-connected, then Cartesian graph bundle \(G\) with fibre \(F\) over the base graph \(B\) is \((p_F + p_B, q_F + q_B)\)-connected. Furthermore, if \(q_F + p_B \geq 0\), then \(G\) is also \((p_F + p_B + 1, q_F + p_B – 1)\)-connected. Finally, let graphs \(G_i\), \(i = 1, \ldots, n\), be \((p_i, q_i)\)-connected and let \(k\) be the number of graphs with \(q_i > 0\). The Cartesian graph product \(G = G_1 \Box G_2 \Box \ldots \Box G_n\) is \((\sum p_i, \sum q_i)\)-connected, and, for \(k \geq 1\), it is also \((\sum p_i + k – 1, \sum q_i – k + 1)\)-connected.

Let \(\gamma_t(D)\) denote the total domination number of a digraph \(D\), and let \(C_m \Box C_n\) denote the Cartesian product graph of \(C_m\) and \(C_n\), where \(C_m\) denotes the directed cycle of length \(m\), \(m \leq n\). In [On domination number of Cartesian product of directed cycles, Information Processing Letters 110 (2010) 171-173], Liu et al. determined the domination number of \(C_2 \Box C_n\), \(C_3 \Box C_n\), and \(C_4 \Box C_n\). In this paper, we determine the exact values of \(\gamma_t(C_m \Box C_n)\) when at least one of \(m\) and \(n\) is even, or \(n\) is odd and \(m = 1, 3, 5,\) or \(7\).

In this paper, a new type of labeled graphs, called modular multiplicative graphs, is introduced and studied. Specifically, we show that every graph is a subgraph of a modular multiplicative graph. Later, we introduce \(k\)-modular multiplicative graphs and prove that certain families of paths and cycles admit such a label. We conclude with several open problems and areas of future possible research, including a note on harmonious graph labels.

Let \(X = (V,E)\) be a digraph. \(X\) is maximally connected if \(\kappa(X) = \delta(X)\). \(X\) is maximally arc-connected if \(\lambda(X) = \delta(X)\). And \(X\) is super arc-connected if every minimum arc-cut of \(X\) is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we prove that the strongly connected Bi-Cayley digraphs are maximally connected and maximally arc-connected, and most strongly connected Bi-Cayley digraphs are super arc-connected.

A \(2\)-rainbow dominating function (2RDF) of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V(G)\) with \(f(v) = \emptyset\), the condition that there exists \(u \in N(v)\) with \(\bigcup_{u\in N(v)}f(u) = \{1,2\}\) is fulfilled, where \(N(v)\) is the open neighborhood of \(v\). A rainbow dominating function \(f\) is said to be a rainbow restrained domination function if the induced subgraph of \(G\) by the vertices with label \(\emptyset\) has no isolated vertex. The weight of a rainbow restrained dominating function is the value \(w(f) = \sum_{u \in V(G)} |f(u)|\). The minimum weight of a rainbow restrained dominating function of \(G\) is called the rainbow restrained domination number of \(G\). In this paper, we initiate the study of the rainbow restrained domination number and we present some bounds for this parameter.

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph and named in honor of Professor Frank Harary. For a connected graph \(G = (V, E)\) with edge connectivity \(\lambda(G) \geq 2\), and an edge \(v_iv_j \in E(G)\), \(G – v_iv_j\) is the subgraph formed from \(G\) by deleting the edge \(v_iv_j\). Denote the Harary index of \(G\) and \(G – v_iv_j\) by \(H(G)\) and \(H(G – v_iv_j)\). Xu and Das [K.X. Xu, K.C. Das, On Harary index of graphs, Discrete Appl. Math. 159 (2011) 1631–1640] obtained lower and upper bounds on \(H(G + v_iv_j) – H(G)\) and characterized the equality cases in those bounds. We find that the equality case in the lower bound is not true and we correct it. In this paper, we give lower and upper bounds on \(H(G) – H(G – v_iv_j)\), and provide some graphs to satisfy the equality cases in these bounds. Furthermore, we extend the Harary index to directed graphs and obtain similar conclusions.

For a connected graph \(G\) of order \(n \geq 2\) and a linear ordering \(s: v_1, v_2, \ldots, v_n\) of \(V(G)\), define \(d(s) = \sum_{i=1}^{n-1} d(v_i, v_{i+1})\). The traceable number \(t(G)\) and upper traceable number \(t^+(G)\) of \(G\) are defined by \(t(G) = \min\{d(s)\}\) and \(t^+(G) = \max\{d(s)\}\), respectively, where the minimum and maximum are taken over all linear orderings \(s\) of \(V(G)\). Consequently, \(t(G) \leq t^+(G)\). It is known that \(n-1 \leq t(G) \leq 2n-4\) and \(n-1 \leq t^+(G) \leq \left\lfloor \frac{n^2}{2} \right\rfloor – 1\) for every connected graph \(G\) of order \(n \geq 3\) and, furthermore, for every pair \(n, A\) of integers with \(2n-1 \leq A \leq 2n-4\) there exists a graph of order \(n\) whose traceable number equals \(A\). In this work, we determine all pairs \(A, B\) of positive integers with \(A \leq B\) that are realizable as the traceable number and upper traceable number, respectively, of some graph. It is also determined for which pairs \(n, B\) of integers with \(n-1 \leq B \leq \left\lfloor \frac{n^2}{2} \right\rfloor – 1\) there exists a graph whose order equals \(n\) and upper traceable number equals \(\mu\).

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. A \(k\)-(p, 1)-total labelling of a graph \(G\) is a function \(f\) from \(V(G) \cup E(G)\) to the color set \(\{0, 1, \ldots, k\}\) such that \(|f(u) – f(v)| \geq 1\) if \(uv \in E(G)\), \(|f(e_1) – f(e_2)| \geq 1\) if \(e_1\) and \(e_2\) are two adjacent edges in \(G\), and \(|f(u) – f(e)| \geq p\) if the vertex \(u\) is incident to the edge \(e\). The minimum \(k\) such that \(G\) has a \(k-(p, 1)\)-total labelling, denoted by \(\lambda_p^T(G)\), is called the \((p, 1)\)-total labelling number of \(G\). In this paper, we prove that, if a 1-planar graph \(G\) satisfies that maximum degree \(\Delta(G) \geq 7p + 1\) and no adjacent triangles in \(G\) or maximum degree \(\Delta(G) \geq 6p + 3\) and no intersecting triangles in \(G\), then \(\lambda_p^T(G) \leq \Delta + 2p – 2\), \(p \geq 2\).

The hyper-star graph \(HS(n,k)\) is defined as follows: its vertex-set is the set of \(\{0, 1\}\)-sequences of length \(n\) with weight \(k\), where the weight of a sequence \(v\) is the number of \(1\)s in \(v\), and two vertices are adjacent if and only if one can be obtained from the other by exchanging the first symbol with a different symbol (\(1\) with \(0\), or \(0\) with \(1\)) in another position. In this paper, we will find the automorphism groups of regular hyper-star and folded hyper-star graphs. Then, we will show that only the graphs \(HS(4, 2)\) and \(FHS(4, 2)\) are Cayley graphs.

Let \(G_1\) and \(G_2\) be two connected graphs. The Kronecker product \(G_1 \times G_2\) has vertex set \(V(G_1 \times G_2) = V(G_1) \times V(G_2)\) and the edge set \(E(G_1 \times G_2) = \{(u_1, v_1), (u_2, v_2) : u_1u_2 \in E(G_1), v_1v_2 \in E(G_2)\}\). In this paper, we show that \(K_n \times K_m\) is super-\(\chi\) for \(n \geq m \geq 2\) and \(n+m \geq 5\), \(K_m \times P_n\) is super-\(\kappa\) for \(n \geq m \geq 3\), and \(K_m \times C_n\) is super-\(\kappa\) for \(n \geq m \geq 3\).

An explicit expression of the restricted edge connectivity of strong product of two triangle-free graphs is presented, which yields a sufficient and necessary condition for these strong product graphs to be super restricted edge connected.

The anti-Ramsey number \(AR(n,G)\), for a graph \(G\) and an integer \(n \geq |V(G)|\), is defined to be the minimal integer \(r\) such that in any edge-colouring of \(K_n\) by at least \(r\) colours there is a multicoloured copy of \(G\), namely, a copy of \(G\) whose edges have distinct colours. In this paper, we determine the anti-Ramsey numbers of all graphs having at most four edges.

In this paper, we determine the unique bicyclic graph with the largest signless Laplacian spectral radius among all the bicyclic graphs with \(n\) vertices and a given girth.

For a connected graph \(G\) and any two vertices \(u\) and \(v\) in \(G\), let \(d(u,v)\) denote the distance between \(u\) and \(v\) and let \(d(G)\) be the diameter of \(G\). For a subset \(S\) of \(V(G)\), the distance between \(v\) and \(S\) is \(d(v, S) = \min\{d(v,x) \mid x \in S\}\). Let \(\Pi = \{S_1, S_2, \ldots, S_k\}\) be an ordered \(k\)-partition of \(V(G)\). The representation of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(r(v \mid \Pi) = (d(v, S_1), d(v, S_2), \ldots, d(v, S_k))\). A partition \(\Pi\) is a resolving partition for \(G\) if the \(k\)-vectors \(r(v \mid \Pi)\), \(v \in V(G)\) are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V(G)\) is the partition dimension of \(G\), and is denoted by \(pd(G)\). A partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) is a resolving path \(k\)-partition for \(G\) if it is a resolving partition and each subgraph induced by \(S_i\), \(1 \leq i \leq k\), is a path. The minimum \(k\) for which there exists a path resolving \(k\)-partition of \(V(G)\) is the path partition dimension of \(G\), denoted by \(ppd(G)\). In this paper, path partition dimensions of trees and the existence of graphs with given path partition, partition, and metric dimension, respectively, are studied.

Let \(A\) be an abelian group with \(|A| \geq 4\). Suppose that \(G\) is a \(3\)-edge-connected simple graph on \(n \geq 19\) vertices. We show in this paper that if \(\max\{d(x), d(y), d(z)\} \geq n/6\) for every \(3\)-independent vertices \(\{x, y, z\}\) of \(G\), then either \(G\) is \(A\)-connected or \(G\) can be \(T\)-reduced to the Petersen graph, which generalizes the result of Zhang and Li (Graphs and Combin., \(30 (2014), 1055-1063).\)

Let \({F}_q\) be a finite field of odd order \(q\). In this note, the generator polynomials and the numbers of all self-dual and self-orthogonal cyclic and negacyclic codes of length \(2^m\) over \({F}_q\) are precisely characterized.

In this paper, we find the star chromatic number \(\chi_s\) for the central graph of sunlet graphs \(C(S_n)\), line graph of sunlet graphs \(L(S_n)\), middle graph of sunlet graphs \(M(S_n)\), and the total graph of sunlet graphs \(T(S_n)\).

Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify the authenticity of the received message. In this paper, we construct multireceiver authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of the two codes are also computed.

For a simple undirected graph \(G\) with vertex set \(V\) and edge set \(E\), a total \(k\)-labeling \(\lambda: V \cup E \rightarrow \{1, 2, \ldots, k\}\) is called a vertex irregular total \(k\)-labeling of \(G\) if for every two distinct vertices \(x\) and \(y\) of \(G\), their weights \(wt(x)\) and \(wt(y)\) are distinct, where the weight of a vertex \(x\) in \(G\) is the sum of the label of \(x\) and the labels of all edges incident with the vertex \(x\). The total vertex irregularity strength of \(G\), denoted by \(\text{tus}(G)\), is the minimum \(k\) for which the graph \(G\) has a vertex irregular total \(k\)-labeling. The complete \(m\)-partite graph on \(n\) vertices in which each part has either \(\left\lfloor \frac{n}{m} \right\rfloor\) or \(\left\lceil \frac{n}{m} \right\rceil\) vertices is denoted by \(T_{n,m}\). The total vertex irregularity strength of some equitable complete \(m\)-partite graphs, namely, \(T_{m,m+1}\), \(T_{m,m+2}\), \(T_{m,2m}\), \(T_{m,2m+4}\), \(T_{3m-1}\) (\(m \geq 4\)), \(T_{n}\) (\(n = 3m+r\), \(r = 1, 2, \ldots, m-1\)), and equitable complete \(3\)-partite graphs have been studied in this paper.

Suppose \(m\) and \(t\) are integers such that \(0 < t \leq m\). An \((m,t)\)-splitting system is a pair \((X, \mathcal{B})\) that satisfies for every \(Y \subseteq X\) with \(|Y| = t\), there is a subset \(B\) of \(X\) in \(\mathcal{B}\), such that \(|B \cap Y| = \left\lfloor \frac{t}{2} \right\rfloor\) or \(|(X \setminus B) \cap Y| = \left\lceil \frac{t}{2} \right\rceil\). Suppose \(m\), \(t_1\), and \(t_2\) are integers such that \(t_1 + t_2 \leq m\). An \((m, t_1, t_2)\)-separating system is a pair \((X, \mathcal{B})\) which satisfies for every \(P \subseteq X\), \(Q \subseteq X\) with \(|P| = t_1\), \(|Q| = t_2\), and \(P \cap Q = \emptyset\), there exists a block \(B \in \mathcal{B}\) for which either \(P \subseteq B\), \(Q \cap B = \emptyset\) or \(Q \subseteq B\), \(P \cap B = \emptyset\). We will give some results on splitting systems and separating systems for \(t = 5\) and \(t = 6\).

Motivated by the recent work by Ramirez \([8]\), related to the bi-periodic Fibonacci sequences, here we introduce the bi-periodic incomplete Lucas sequences that gives the incomplete Lucas sequence as a special case. We also give recurrence relations and the generating function of these sequences. Also, we give a relation between bi-periodic incomplete Fibonacci sequences and bi-periodic incomplete Lucas sequences.

In this paper, we prove the \(q\)-log-convexity of Domb’s polynomials, which was conjectured by Sun in the study of series for powers of \(\pi\). As a result, we obtain the log-convexity of Domb’s numbers. Our proof is based on the \(q\)-log-convexity of Narayana polynomials of type \(B\) and a criterion for determining \(q\)-log-convexity of self-reciprocal polynomials.

Two Schwenk-like formulas about the signless Laplacian matrix of a graph are given, and thus it gives new tools for computing \(Q\)-
characteristic polynomials of graphs directly. As an application, we give the \(Q\)-characteristic polynomial of lollipop graphs and reprove the known result that no two non-isomorphic lollipop graphs are \(Q\)-cospectral by a simple manner.

In this paper, we give a general result which enlarge the class of graphs known to have \(\alpha\)-labeling.

An independent set in a graph \(G\) is a subset \(I\) of the vertices such that no two vertices in \(I\) are adjacent. We say that \(I\) is a maximum independent set in \(G\) if no other independent set is larger than \(I\). In this paper, we study the problem of determining the second and third largest number of maximum independent sets among all trees and forests. Extremal graphs achieving these values are also given.

This paper is motivated by the concept of the signed \(k\)-independence problem and dedicated to the complexity of the problem on graphs. We show that the problem is linear-time solvable for any strongly chordal graph with a strong elimination ordering and polynomial-time solvable for distance-hereditary graphs. For any fixed positive integer \(k \geq 1\), we show that the signed \(k\)-independence problem on chordal graphs and bipartite planar graphs is NP-complete. Furthermore, we show that even when restricted to chordal graphs or bipartite planar graphs, the signed \(k\)-independence problem, parameterized by a positive integer \(k\) and weight \(\kappa\), is not fixed-parameter tractable.

Edge minimal Hamilton laceable bigraphs on \(2m\) vertices have at least \(\left\lfloor \frac{m+3}{6} \right\rfloor\) vertices of degree \(2\). If a bigraph is edge minimal with respect to Hamilton laceability, it is by definition edge critical, meaning the deletion of any edge will cause it to no longer be Hamilton laceable. The converse need not be true. The \(m\)-crossed prisms \([8]\) on \(4m\) vertices are edge critical for \(m \geq 2\) but not edge minimal since they are cubic. A simple modification of \(m\)-crossed prisms forms a family of “sausage” bigraphs on \(4m + 2\) vertices that are also cubic and edge critical. Both these families share the unusual property that they have exponentially many Hamilton paths between every pair of vertices in different parts. Even so, since the bigraphs are edge critical, deleting an arbitrary edge results in at least one pair having none.

In this paper, we investigate some new identities of symmetry for the Carlitz \(q\)-Bernoulli polynomials invariant under \(S_4\), which are derived from \(p\)-adic \(q\)-integrals on \(\mathbb{Z}_p\).

In this paper, we give the definition of acyclic total coloring and acyclic total chromatic number of a graph. It is proved that the acyclic total chromatic number of a planar graph \(G\) with maximum degree \(\Delta(G)\) and girth \(g\) is at most \(\Delta(G)+2\) if \(\Delta \geq 12\), or \(\Delta \geq 6\) and \(g \geq 4\), or \(\Delta = 5\) and \(g \geq 5\), or \(g \geq 6\). Moreover, if \(G\) is a series-parallel graph with \(\Delta \geq 3\) or a planar graph with \(\Delta \geq 3\) and \(g \geq 12\), then the acyclic total chromatic number of \(G\) is \(\Delta(G) + 1\).