A construction of authentication codes with arbitration from singular symplectic geometry over finite fields is given, and the parameters of the codes are computed. Assuming that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the probabilities of success for different types of deceptions are also computed.

Let \(M\) be a simple connected binary matroid with corank at least two such that \(M\) has no connected hyperplane. Seymour proved that \(M\) has a non-trivial series class. We improve this result by proving that \(M\) has at least two disjoint non-trivial series classes \(L_1\) and \(L_2\) such that both \(M \backslash L_1\) and \(M \backslash L_2\) are connected. Our result extends the corresponding result of Kriesell regarding critically \(2\)-connected graphs.

For a non-complete graph \(\Gamma\), a vertex triple \((u,v,w)\) with \(v\) adjacent to both \(u\) and \(w\) is called a \(2\)-geodesic if \(u \neq w\) and \(u,w\) are not adjacent. Then \(\Gamma\) is said to be \(2\)-geodesic transitive if its automorphism group is transitive on both arcs and \(2\)-geodesics. In this paper, we classify the family of connected \(2\)-geodesic transitive graphs of valency \(3p\), where \(p\) is an odd prime.

We generalize the well known congruence Lucas\(^1\) Theorem for binomial coefficient to the bi\(^s\)nomial coefficients.

The linear arboricity \(la(G)\) of a graph \(G\) is the minimum number of linear forests that partition the edges of \(G\). In this paper, it is proved that if \(G\) is a planar graph with maximum degree \(\Delta \geq 7\) and every \(7\)-cycle of \(G\) contains at most two chords, then \(la(G) = \left\lceil \frac{\Delta(G)}{2} \right\rceil\).

In this paper, we study the generalized Pell \(p\)-sequences modulo \(m\). Additionally, we define the generalized Pell \(p\)-sequences and the basic generalized Pell sequences in groups, and then examine these sequences in finite groups. Furthermore, we obtain the periods of the generalized Pell \(p\)-sequences and the basic periods of the basic generalized Pell sequences in the binary polyhedral groups \(\langle n,2,2\rangle\), \(\langle2,n,2\rangle\), and \(\langle2,2,n\rangle\).

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined as the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the star graphs, one of the most popular interconnection networks.

The terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arose in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees that attain these bounds. Additionally, we investigate the properties of extremal trees that attain the maximum terminal Wiener index among all trees of order \(n\) with fixed maximum degree.

Based on some results of Shult and Yanushka [7], Brouwer [1] proved that there exists a unique regular near hexagon with parameters \((s,t,t_2) = (2,11,1)\), namely the one related to the extended ternary Golay code. His proof relies on the uniqueness of the Witt design \(S(5,6,12)\), Pless’s characterization of the extended ternary Golay code \(G_{12}\), and some properties of \(S(5,6,12)\) and \(G_{12}\). It is possible to avoid all this machinery and provide an alternative, more elementary and self-contained proof for the uniqueness. The author recently observed that such an alternative proof is implicit in the literature, obtainable by combining results from [1], [4], and [7]. This survey paper aims to bring this fact to the attention of the mathematical community. We describe the relevant parts of the above papers for this alternative proof of classification. Additionally, we prove several extra facts not explicitly contained in [1], [4], or [7]. This paper can also be seen as an addendum to Section 6.5 of [3], where the uniqueness of the near hexagon was not proved.

Recently, Belbachir and Belkhir gave some recurrence relations for the \(r\)-Lah numbers. In this paper, we give other properties for the \(r\)-Lah numbers, we introduce and study a restricted class of these numbers.

An \(H\)-polygon is a simple polygon whose vertices are \(H\)-points, which are points of the set of vertices of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. Let \(G(v)\) denote the least possible number of \(H\)-points in the interior of a convex \(H\)-polygon \(K\) with \(v\) vertices. In this paper, we prove that \(G(8) = 2\), \(G(9) = 4\), \(G(10) = 6\), and \(G(v) \geq \lceil \frac{v^2}{16\pi^2}-\frac{v}{4}+\frac{1}{2}\rceil – 1\) for all \(v \geq 11\), where \(\lceil x \rceil\) denotes the minimal integer more than or equal to \(x\).

Row-cyclic array codes have already been introduced by the author \([9]\). In this paper, we give some special classes of row-cyclic array codes as an extension of classical BCH and Reed-Solomon codes.

The harmonic weight of an edge is defined as reciprocal of the average degree of its end-vertices. The harmonic index of a graph \(G\) is defined as the sum of all harmonic weights of its edges. In this work, we give the minimum value of the harmonic index for any \(n\)-vertex connected graphs with minimum degree \(\delta\) at least \(k(\geq n/2)\) and show the corresponding extremal graphs have only two degrees,i.e., degree \(k\)and degree \(n – 1\), and the number of vertices of degree \(k\) is as close to \(n/2\) as possible.

In this note, we consider one type of \(k\)-tridiagonal matrix family whose permanents and determinants are specified to the balancing and Lucas-balancing numbers. Moreover, we provide some properties between Chebyshev polynomial properties and the given number
sequences,

Let \(G = (V, E)\) be a graph. A subset \(D \subseteq V\) is a dominating set if every vertex not in \(D\) is adjacent to a vertex in \(D\). The domination number of \(G\) is the smallest cardinality of a dominating set of \(G\). The bondage number of a nonempty graph \(G\) is the smallest number of edges whose removal from \(G\) results in a graph with larger domination number than \(G\). In this paper, we determine that the exact value of the bondage number of an \((n-3)\)-regular graph \(G\) of order \(n\) is \(n-3\).

A graph is closed when its vertices have a labeling by \([n]\) with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.

Let \(\Sigma = (X, \mathcal{B})\) be a \(4\)-cycle system of order \(v = 1 + 8k\). A \(c\)-colouring of type \(s\) is a map \(\phi: \mathcal{B} \to C\), where \(C\) is a set of colours, such that exactly \(c\) colours are used and for every vertex \(x\), all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(4k = qs + r\), with \(r \geq 0\). \(\phi\) is equitable if for every vertex \(x\), the set of the \(4k\) blocks containing \(x\) is partitioned into \(r\) colour classes of cardinality \(q + 1\) and \(s – r\) colour classes of cardinality \(q\). In this paper, we study colourings for which \(s | k\), providing a description of equitable block colourings for \(c \in \{s, s+1, \ldots, \lfloor \frac{2s^2+s}{3} \rfloor\}\).

In this paper, we first introduce a linear program on graphical invariants of a graph \(G\). As an application, we attain the extremal graphs with lower bounds on the first Zagreb index \(M_1(G)\), the second Zagreb index \(M_2(G)\), their multiplicative versions \(\Pi_1^*(G)\), \(\Pi_2(G)\), and the atom-bond connectivity index \(ABC(G)\), respectively.

Let \(\Gamma\) be an oriented graph. We denote the in-neighborhood and out-neighborhood of a vertex \(v\) in \(\Gamma\) by \(\Gamma^-(v)\) and \(\Gamma^+(v)\), respectively. We say \(\Gamma\) has Property \(A\) if, for each arc \((u,v)\) in \(\Gamma\), each of the graphs induced by \(\Gamma^+(u) \cap \Gamma^+(v)\), \(\Gamma^-(u) \cap \Gamma^-(v)\), \(\Gamma^-(u) \cap \Gamma^+(v)\), and \(\Gamma^+(u) \cap \Gamma^-(v)\) contains a directed cycle. Moreover, \(\Gamma\) has Property B if each arc \((u,v)\) in \(\Gamma\) extends to a \(3\)-path \((x,u), (u,v), (v,w)\), such that \(|\Gamma^+(x) \cap \Gamma^+(u)| \geq 5\) and \(|\Gamma^-(v) \cap \Gamma^-(w)| \geq 5\). We show that the only oriented graphs of order at most \(17\), which have both properties \(A\) and \(B\), are the Tromp graph \(T_{16}\) and the graph \(T^+_{16}\), obtained by duplicating a vertex of \(T_{16}\). We apply this result to prove the existence of an oriented planar graph with oriented chromatic number at least \(18\).

By the partial fraction decomposition method, we establish a \(q\)-harmonic sum identity with multi-binomial coefficient, from which we can derive a fair number of harmonic number identities.

A fall \(k\)-coloring of a graph \(G\) is a proper \(k\)-coloring of \(G\) such that each vertex of \(G\) sees all \(k\) colors on its closed neighborhood. We denote \(\text{Fall}(G)\) the set of all positive integers \(k\) for which \(G\) has a fall \(k\)-coloring. In this paper, we study fall colorings of the lexicographic product of graphs and the categorical product of graphs. Additionally, we show that for each graph \(G\), \(\text{Fall}(M(G)) = \emptyset\), where \(M(G)\) is the Mycielskian of the graph \(G\). Finally, we prove that for each bipartite graph \(G\), \(\text{Fall}(G^c) \subseteq \{\chi(G^c)\}\) and it is polynomial time to decide whether \(\text{Fall}(G^c) = \{\chi(G^c)\}\) or not.

In this paper, we first provide two necessary conditions for a graph \(G \) to be \(E_k\)-cordial, then we prove that every \(P_n(n \geq 3)\) is \(E_p\)-cordial if \(p\) is odd. In the end, we discuss the \(E_2\)-cordiality of a graph \)G\) under the condition that some subgraph of \(G\) has a \(1\)-factor.

In this paper, we consider the problem of determining the structure of a minimal critical set in a latin square \(L\) representing the elementary abelian \(2\)-group of order \(8\).

In this paper, the first two (resp. four) largest signless Laplacian spectral radii together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order n are determined, and the first two (resp. four) largest signless Laplacian spreads together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order \(n\) are identified.

An edge-magic total labeling of a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(\{1, 2, \ldots, |V(G) \cup E(G)|\}\) with the property that there exists an integer constant \(c\) such that \(\lambda(x) + \lambda(x,y) + \lambda(y) = c\) for any \((x, y) \in E(G)\). If \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then the edge-magic total labeling is called super edge-magic total labeling. In this paper, we formulate super edge-magic total labeling on subdivisions of stars \(K_{1,p}\), for \(p \geq 5\).

In this paper, we briefly survey Euler’s works on identities connected with his famous Pentagonal Number Theorem. We state a partial generalization of his theorem for partitions with no part exceeding an identified value \(k\), along with some identities linking total partitions to partitions with distinct parts under the above constraint. We derive both recurrence formulas and explicit forms for \(\Delta_n(m)\), where \(\Delta_n(m)\) denotes the number of partitions of \(m\) into an even number of distinct parts not exceeding \(n\), minus the number of partitions of \(m\) into an odd number of distinct parts not exceeding \(n\). In fact, Euler’s Pentagonal Number Theorem asserts that for \(m \leq n\), \(\Delta_n(m) = \pm 1\) if \(m\) is a Pentagonal Number and \(0\) otherwise. Finally, we establish two identities concerning the sum of bounded partitions and their connection to prime factors of the bound integer.

Consider the following one-person game: let \(S = {F_1, F_2,\ldots, F_r}\) be a family of forbidden graphs. The edges of a complete graph are randomly shown to the Painter one by one, and he must color each edge with one of \(r\) colors when it is presented, without creating some fixed monochromatic forbidden graph \(F\); in the \(i\)-th color. The case of all graphs \(F\); being cycles is studied in this paper. We give a lower bound on the threshold function for online \(S\)-avoidance game,which generalizes the results of Marciniszyn, Spdhel and Steger for the symmetric case. [Combinatorics, Probability and Computing, Vol. \(18, 2009: 271-300.\)]

Given positive integers \(n\), \(k\), and \(m\), the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind is the number of permutations of an \(n\)-set with \(k\) disjoint cycles of length \(\leq m\). By inverting the matrix consisting of the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind as the \((n+1,k+1)\)-th entry, the \((n,k)\)-th \(m\)-restrained Stirling number of the second kind is defined. In this paper, we study the multi-restrained Stirling numbers of the first and second kinds to derive their explicit formulae, recurrence relations, and generating functions. Additionally, we introduce a unique expansion of multi-restrained Stirling numbers for all integers \(n\) and \(k\), and a new generating function for the Stirling numbers of the first kind.

Employing \(q\)-commutive structures, we develop binomial analysis and combinatorial applications induced by an important operator in
analogue Fourier analysis associated with well-known \(q\)-series of L.J. Rogers.

In [H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance
pooling designs, Discrete Math. \(243 (2002) 167-170\)], by using subspaces
in a vector space Ngo and Du constructed a family of well-known pooling
designs. In this paper, we construct a family of pooling designs by using
bilinear forms on subspaces in a vector space, and show that our design and
Ngo-Du’s design have the same error-tolerance capability but our design is
more economical than Ngo-Du’s design under some conditions.

A transverse Steiner quadruple system \((TSQS)\) is a triple \((X, \mathcal{H}, \mathcal{B})\) where \(X\) is a \(v\)-element set of points, \(\mathcal{H} = \{H_1, H_2, \ldots, H_r\}\) is a partition of \(X\) into holes, and \(\mathcal{B}\) is a collection of transverse \(4\)-element subsets with respect to \(\mathcal{H}\), called blocks, such that every transverse \(3\)-element subset is in exactly one block. In this article, we study transverse Steiner quadruple systems with \(r\) holes of size \(g\) and \(1\) hole of size \(u\). Constructions based on the use of \(s\)-fans are given, including a construction for quadrupling the number of holes of size \(g\). New results on systems with \(6\) and \(11\) holes are obtained, and constructions for \(\text{TSQS}(x^n(2n)^1)\) and \(\text{TSQS}(4^n2^1)\) are provided.

Let \(G\) be a subgraph of the complete graph \(K_{r+1}\) on \(r+1\) vertices, and let \(K_{r+1} – E(G)\) be the graph obtained from \(K_{r+1}\) by deleting all edges of \(G\). A non-increasing sequence \(\pi = (d_1, d_2, \ldots, d_n)\) of nonnegative integers is said to be potentially \(K_{r+1} – E(G)\)-graphic if it is realizable by a graph on \(n\) vertices containing \(K_{r+1} – E(G)\) as a subgraph. In this paper, we give characterizations for \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{r+1} – E(G)\)-graphic for \(G = 3K_2, K_3, P_3, K_{1,3}\), and \(K_2 \cup P_2\), which are analogous to Erdős-Gallai’s characterization using a system of inequalities. These characterizations partially answer one problem due to Lai and Hu [10].

An unoriented flow in a graph is an assignment of real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. This is equivalent to a flow in a bidirected graph where all edges are extraverted. A nowhere-zero unoriented \(k\)-flow is an unoriented flow with values from the set \(\{\pm 1, \ldots, \pm( k-1)\}\). It has been conjectured that if a graph admits a nowhere-zero unoriented flow, then it also admits a nowhere-zero unoriented \(6\)-flow. We prove that this conjecture holds true for Hamiltonian graphs, with \(6\) replaced by \(12\).

Let \(G\) be a graph with vertex set \(V(G)\), \(d_G(u,v)\) and \(\delta_G(v)\) denoteas the topological distance between vertices \(u\) and \(v\) in \(G\), and \(d_G(v)\) as the degree of vertex \(v\) in \(G\),respectively. The Schultz polynomial of \(G\) is defined as \(H^+(G) = \sum\limits_{u,v \subseteq V(G)} (\delta _G(u)+\delta _G(v))x^{d_G(u,v)}\), and the modified Schultz polynomial of \(G\) is defined as \(H^*(G) = \sum\limits_{u,v \subseteq V(G)}(\delta _G(u)+\delta _G(v)) x^{d_G(u,v)}\). In this paper, we obtain explicit analytical expressions for the expected values of the Schultz polynomial and modified Schultz polynomial of a random benzenoid chain with $n$ hexagons. Furthermore, we derive expected values of some related topological indices.

For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\). The orientation number of \(G\), denoted by \(\vec{d}(G)\), is defined as \(\min\{d(D) \mid D \in \mathcal{D}(G)\}\), where \(d(D)\) denotes the diameter of the digraph \(D\). In this paper, we prove that \(\vec{d}(P_3 \times K_5) = 4\) and \(\vec{d}(C_8 \times K_3) = 6\), where \(\times\) is the tensor product of graphs.

In this paper, we consider the domination number, the total domination number, the restrained domination number, the total restrained domination number and the strongly connected domination number of lexicographic product digraphs.

Radio labeling is a variation of Hale’s channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph \(G\) subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph \(G\) is a function \(c: V(G) \to \mathbb{Z}_+\) such that \[d(u, v) + |c(u) – c(v)| \geq 1 + \text{diam}(G)\] for every two distinct vertices \(u\) and \(v\) of \(G\), where \(d(u, v)\) is the distance between \(u\) and \(v\). The \emph{span} of a radio labeling is the maximum integer assigned to a vertex. The \emph{radio number} of a graph \(G\) is the minimum span, taken over all radio labelings of \(G\). This paper establishes the radio number of the Cartesian product of a cycle graph with itself,( i.e., of \(C_n \Box C_n\)).

In this note we present an application of \(q\)-Lucas theorem, from which the \(q\)-binomial rational root theorem obtained by K. R. Slavin can be deduced as a special case.