Let \(G\) be a finite abelian group. The critical number \(cr(G)\) of \(G\) is the least positive integer \(m\) such that every subset \(A \subseteq G \setminus \{0\}\) of cardinality at least \(m\) spans \(G\), i.e., every element of \(G\) can be expressed as a nonempty sum of distinct elements of \(A\). Although the exact values of \(cr(G)\) have been recently determined for all finite abelian groups, the structure of subsets of cardinality \(cr(G) – 1\) that fail to span \(G\) remains characterized except when \(|G|\) is even or \(|G| = pq\) with \(p, q\) primes. In this paper, we characterize these extremal subsets for \(|G| \geq 36\) and \(|G|\) even, or \(|G| = pq\) with \(p, q\) primes and \(q \geq 2p + 3\).

In this paper, we give a criterion to judge whether a linear code over the ring is self-dual. Moreover, we introduce the generating set in standard form for the cyclic codes over \(F_p + vF_p\), and characterize the structure of cyclic codes over the ring. Then we prove that cyclic codes over the ring are principally generated and obtain the unique generating idempotent for cyclic codes of length \(n\), where \(n\) is coprime to \(p\).

Let \(G\) be a finite abelian group, and let \(S\) be a sequence over \(G\). For a sequence \(S\), denote by \(f(S)\) the number of elements in \(G\) that can be expressed as the sum of a nonempty subsequence of \(S\). In this paper, we determine all sequences \(S\) that contain no zero-sum subsequences and satisfy \(f(S) \leq 2|S| – 1\).

For given a graph \(H\), agraphic sequence \(\pi = (d_1, d_2,\ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(m\) containing \(H\) asa subgraph. Let \(K_m- H\) be the graph obtained from \(K_m\), by removing the edges set \(E(H)\) where \(H\) is a subgraph of \(K_m\). In this paper, we characterize potentially \(K_{2,5}\)-graphic sequences. This characterization implies a special case of a theorem due to Yin \(et \;al. [26]\).

The harmonic index of a graph \(G\) is defined as the sum of weights Tay raey of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\), respectively. In this paper, we give a sharp lower bound on the harmonic index of trees with a perfect matching in terms of the number of vertices. A sharp lower bound on the harmonic index of trees with a given size of matching is also obtained.

Graphs \(S[n,k]\) are introduced as the graphs obtained from the Sierpiński graphs \(S(n, k)\) by contracting edges that lie in no complete subgraph \(K_k\). The family \(S[n,k]\) generalizes the previously studied class of Sierpiński gasket graphs \(S_k\). We investigate various properties of graphs \(S[n,k]\), particularly focusing on hamiltonicity and chromatic number.

Let \(G\) be a graph. The Randić index of \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) denote the degrees of vertices \(u\) and \(v\) in \(G\). In this paper, we establish a sharp upper bound for the Randić index \(R(G)\) among all unicyclic graphs \(G\) with \(n\) vertices, \(k\) pendant vertices, and \(n \geq 3k\), where \(k \geq 3\).

Let \(G\) be a simple quadrangulation on a closed surface \(F^2\). Two reductions for quadrangulations are defined in this paper: face-contraction and \(4\)-cycle removal. We define four types of irreducibility:

1. \(G\) is \({irreducible}\) if any face-contraction breaks the simplicity of \(G\).
2. \(G\) is \(\mathcal{D}_3\)-\({irreducible}\) if \(G\) has minimum degree at least 3 and any face-contraction or 4-cycle removal breaks simplicity or reduces minimum degree to less than 3.
3. \(G\) is \(\mathcal{K}_3\)\({-irreducible}\) if \(G\) is 3-connected and any face-contraction or 4-cycle removal breaks simplicity or 3-connectedness.
4. \(G\) is \(\mathcal{S}_4\) \({-irreducible}\) if \(G\) has no separating 4-cycle and any face-contraction breaks simplicity or creates a separating 4-cycle.

In [7] that, except for the sphere and projective plane, irreducibility and \(\mathcal{D}_3\)-irreducibility of quadrangulations are equivalent. In this paper, we prove that for all surfaces, \(\mathcal{D}_3\)-irreducibility and \(\mathcal{K}_3\)-irreducibility are equivalent. Additionally, we prove that for the sphere, projective plane, and torus, \(\mathcal{D}_3\)-irreducibility and \(\mathcal{S}_4\)-irreducibility are equivalent, but this does not hold for surfaces of high genus.

An adjacent vertex-distinguishing edge coloring ,avd-coloring for short, of a graph \(G\) is a proper edge coloring of \(G\) such that no pair of adjacent vertices are incident to the same set of colors. We denote the avd-chromatic number of \(G\) by \(\chi’_{avd}(G)\), which is the smallest integer \(k\) such that \(G\) has an avd-coloring with \(k\) colors, and the maximum degree of \(G\) by \(\Delta(G)\). In this paper, we prove that \(\chi’_{avd}(G) \leq \Delta(G) + 4\) for every planar graph \(G\) without isolated edges whose girth is at least five. Notably, this bound is nearly sharp, as \(\chi’_{avd}(C_5) = \Delta(C_5) + 3\).

Let \(G\) be a simple graph of order \(n\). A dominating set of \(G\) is a set \(S\) of vertices of \(G\) such that every vertex of \(G\) is either in \(S\) or adjacent to a vertex in \(S\). The domination polynomial of \(G\) is defined as \(D(G, x) = \sum_{i=0}^{n} d(G, i)x^i\), where \(d(G, i)\) denotes the number of dominating sets of \(G\) of size \(i\). In this paper, we demonstrate that cycles are uniquely determined by their domination polynomials.

The third-order Randić index of a graph \(G\) is defined as \(R_s(G) = \sum_{u_1u_2u_3u_4} \frac{1}{\sqrt{d(u_1) d(u_2) d(u_3) d(u_4)}}\), where the summation is taken over all possible paths of length three in \(G\). In this paper, we first derive a recursive formula for computing the third-order Randić index of a double hexagonal chain. Furthermore, we establish upper and lower bounds for the third-order Randić index and characterize the double hexagonal chains that achieve the extremal third-order Randić index.

The decycling index of a digraph \(D\) is defined to be the minimum number of arcs in a set whose removal from \(D\) leaves an acyclic digraph. In this paper, we obtain some results on the decycling index of bipartite tournaments.

In this paper two authentication codes with multiple arbiters are constructed to protect the communication system against the attacks from the opponent, transmitter, receiver and dishonest arbiters. The first construction takes advantage of set theory to give an authentication codes with two arbiters that resists collusion attacks from dishonest arbiters and participators availably. The second construction makes full use of of Reed- Solomon-code (\(RS\)-code) and \((k, n)\)-threshold scheme to give an authentication codes with \(n\) arbiters that effectively prevents multiple arbiters from cheating.

A directed Toeplitz graph is a digraph with a Toeplitz adjacency matrix. In this paper we contribute to [6]. The paper [6] investigates the hamiltonicity of the directed Toeplitz graphs \(T_n\langle s_1,s_2,…, s_k;t_1, t_2,…,t_l\rangle\) with \(s_2 = 2\) and in particular those with \(s_3 = 3\). In this paper we extend this investigation to \(s_2 = 3\) with \(s_1 =t_1 =1\).

W. Y. C. Chen and R. P. Stanley have characterized the symmetries of the \(n\)-cube that act as derangements on the set of \(k\)-faces. In this paper we aim to use their result to characterize those finite subgroups of symmetries whose non-trivial members are derangements of the set of \(k\)-faces.

A sequential labeling of a simple graph G (non-tree) with m edges is an injective labeling f such that the vertex labels \(f(x)\) are from \({0,1,…,m-1}\) and the edge labels induced by \(f(x) + f(y)\) for each edge \(xy\) are distinct consecutive positive integers. A graph is sequential if it has a sequential labeling. We give some properties of sequential labeling and the criterion to verify sequential labeling. Necessary and sufficient conditions are obtained for every case of sequential graphs. A complete characterization of non-tree sequential graphs is obtained by vertex closure. Also, characterizations of sequential trees are given. The structure of sequential graphs is revealed.

In this paper, we give explicit algorithms to compute generating functions of some special sequences, based on the operations of differential operators and shift operators in the non-commutative context and Zeilberger’s holonomic algorithm.
It can be found that not only ordinary generating functions and exponential generating functions but also generating functions of the general form \(\sum_{n} a_n(x)w(y, n)\) can now be computed automatically. Moreover, we generalize this approach and present explicit algorithms to compute \(2\)-variable ordinary power series generating functions and mixed-type generating functions. As applications, various examples are given in the paper.

The graphs we consider are all countable. A graph \(U\) is universal in a given set \(\mathcal{P}\) of graphs if every graph in \(\mathcal{P}\) is an induced subgraph of \(U\) and \(U \in \mathcal{P}\). In this paper we show the existence of a universal graph in the set of all countable graphs with block order bounded by a fixed positive integer. We also investigate some classes of interval graphs and work towards finding universal graphs for them. The sets of graphs we consider are all examples of induced-hereditary graph properties.

In this paper, we give the Hahn polynomials represents by Carlitz’s \(q\)-operators, then show how to deduce Carlitz type generating functions by the technique of exponential operator decomposition.

The Wiener polarity index of a graph \(G\), denoted by \(W_p(G)\), is the number of unordered pairs of vertices \(u, v\) such that the distance between \(u\) and \(v\) is three, introduced by Harold Wiener in 1947. This index is utilized to demonstrate quantitative structure-property relationships in various acyclic and cyclic hydrocarbons. In this paper, we investigate the Wiener polarity index on the Cartesian, direct, strong, and lexicographic products of two non-trivial connected graphs.

Given a graph \(G := (V, E)\) and an integer \(k \geq 2\), the \({component \;order\; edge connectivity}\) of \(G\) is the smallest size of an edge set \(D\) such that the subgraph induced by \(G – D\) has all components of order less than \(k\). Let \({G}(n,m)\) denote the collection of simple graphs \(G\) with \(n\) vertices and \(m\) edges. In this paper, we investigate properties of component order edge connectivity for \({G}(n,m)\), particularly proving results on the maximum and minimum values of this connectivity measure for \({G}(n,m)\) specific values of \(n\), \(m\), and \(k\).

Let \(D\) be a simple digraph without loops and parallel arcs. Deng and Kelmans [A. Deng, A. Kelmans, Spectra of digraph transformations, Linear Algebra and its Applications, \(439(2013) 106-132]\) gave the definition of transformation digraphs by introducing symbol \(‘0’\) and \(‘1’\), and investigated the regular and spectra of digraph transformation. In this paper we discuss a class of total transformation digraphs associate with symbol \(‘0’\). Furthermore, we determine the regularity of these ten new kinds of total transformation digraphs and also give necessary and sufficient conditions for them to be strongly connected.

In this paper, using the generating function, we derive Binet formulas and determinant expressions for the k-generalized Fibonacci numbers and Lucas numbers. As applications, we obtain some new recurrence relations for the Stirling numbers of the second kind and power sums.

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. Let \(p\) be a prime. In [J. Combin. Theory B \(97 (2007) 627-646]\), Feng and Kwak classified connected cubic symmetric graphs of order \(4p\) or \(4p^2\). In this article, all connected cubic symmetric graphs of order \(4p^2\) are classified. It is shown that up to isomorphism there is one and only one connected cubic symmetric graph of order \(4p^3\) for each prime \(p\), and all such graphs are normal Cayley graphs on some groups.

An edge-magic total \((EMT)\) labeling on a graph \(G\) is
a one-to-one mapping \(\lambda : V(G) \cup E(G) \to {1,2,—,|V(G)| +
|E(G)|}\) such that the set of edge weights is one point set, i.e. for
any edge \(xy \in G, w(xy) = {a}\) where \(a = \lambda(x) + \lambda(y) + \lambda(xy)\)
is called a magic constant. If \(\lambda(V(G)) = {1,2,—,|V(G|}\) then an
edge-magic total labeling is called a super edge-magic total labeling.
In this paper, we formulate a super edge-magic total labeling for
a particular tree family called subdivided star \(T(l_1,l_2,\ldots,l_p)\) for
\(p>3\).

Let \(G\) be an edge-colored graphs. A heterochromatic path of \(G\) is such a path in which no two edges have the same color. Let \(g^c(G)\) and \(d^c(v)\) denote the heterochromatic girth and the color degree of a vertex \(v\) of \(G\), respectively. In this paper, some color degree and heterochromatic girth conditions for the existence of heterochromatic paths are obtained.

Let \(\mathcal{U}_m^{W}\) denote the set of unicyclic weighted graphs of size \(m\) with weight \(W\). In this paper, we determine the weighted graph in \(\mathcal{U}_m^{W}\) with maximum spectral radius.

A subset of vertices of a graph \(G\) is called a feedback vertex set of \(G\) if its removal results in an acyclic subgraph. In this paper, we investigate the feedback vertex set of generalized Kautz digraphs \(GK(2,n)\). Let \(f(2,n)\) denote the minimum cardinality over all feedback vertex sets of the generalized Kautz digraph \(GK(2,n)\). We obtain the upper bound of \(f(2,n)\) as follows:
\[f(2,n) \leq n-(\left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{{n-2}}{3} \right\rfloor + \lfloor \frac{n-8}{9}\rfloor)\].

Let \(G\) be a graph of order \(n\) and let \(\mu\) be an eigenvalue of multiplicity \(m\). A star complement for \(\mu\) in \(G\) is an induced subgraph of \(G\) of order \(n-m\) with no eigenvalue \(\mu\). In this paper, we investigate maximal and regular graphs that have \(K_{r,s} + t{K_{1}}\) as a star complement for \(\mu\) as the second largest eigenvalue. Interestingly, it turns out that some well-known strongly regular graphs are uniquely determined by such a star complement.

Given a graph \(G\) and a non-negative integer \(g\), the \(g\)-extra-connectivity of \(G\), denoted by \(\kappa_g(G)\), is the minimum cardinality of a set of vertices of \(G\), if any, whose deletion disconnects \(G\) and every remaining component has more than \(g\) vertices. Note that \(\kappa_0(G)\) and \(\kappa_1(G)\) correspond to the usual connectivity and restricted vertex connectivity of \(G\), respectively. In this paper, we determine \(\kappa_g(FQ_n)\) for \(0 \leq g \leq n-4\), \(n \geq 8\), where \(FQ_n\) denotes the \(n\)-dimensional folded hypercube.

The construction of association schemes based on the subspaces of type \((2,0,1)\) in singular symplectic space over finite fields is provided in this paper.Applying the matrix method and combinatorial design theory, all parameters of the association scheme are computed.

The number of colors required to properly color the edges of a simple graph \(G\) such that any two vertices are incident with different sets of colors is referred to as the vertex-distinguishing edge chromatic number, denoted by \(\chi’_{vd}(G)\). This paper explores an interesting phenomenon concerning vertex-distinguishing proper edge coloring. Specifically, we prove that for every integer \(m \geq 3\), there exists a graph \(G\) of maximum degree \(m\) with \(\chi’_{vd}(G) < \chi'_{vd}(H)\), where \(H\) is a proper subgraph of \(G\).

We evaluate the convolution sums

\(\sum_{l+30m=n} \sigma(l) \sigma(m), \sum_{3l+10m=n} \sigma(l) \sigma(m),\\ \)
\(\sum_{2l+15m=m} \sigma(l) \sigma(m), \sum_{5l+6m=n} \sigma(l) \sigma(m), \\\)
\(\sum_{l+33m=n} \sigma(l) \sigma(m), \sum_{3l+11m=n} \sigma(l) \sigma(m), \\\)
\(\sum_{l+39m=n} \sigma(l) \sigma(m), \sum_{3l+13m=n} \sigma(l) \sigma(m),\\\)

for all \(n \in \mathbb{N}\) using the theory of quasimodular forms, and utilize these convolution sums to determine the number of representations of a positive integer \(n\) by the forms
\[x_1^2 +x_1x_2+ x_2^2 + x_3^2 +x_3x_4+ x_4^2
+ a(x_5^2 + x_5x_6+x_6^2 + x_7^2 + x_7x_8+x_8^2), \]
for \(a = 10, 11, 13\). Quasimodular forms, divisor functions, convolution sums, representation number \(11A25,11F11,11F25,11F20\)

For positive integers \(r\) and \(k_1, k_2, \ldots, k_r\), the van der Waerden number \(W(k_1, k_2, \ldots, k_r; r)\) is the minimum integer \(N\) such that whenever the set \(\{1, 2, \ldots, N\}\) is partitioned into \(r\) sets \(S_1, S_2, \ldots, S_r\), there exists a \(k_i\)-term arithmetic progression contained in \(S_i\) for some \(i\). This paper establishes an asymptotic lower bound for \(W(k, m; 2)\) for fixed \(m \geq 3\), improving upon the result of T.C. Brown et al. in [Bounds on some van der Waerden numbers.J. Combin. Theory, Ser.A \(115 (2008), 1304-1309]\). Additionally, we propose lower bounds on certain van der Waerden-like functions.

The chromatic sum \(\Sigma(G)\) of a graph \(G\) is the smallest sum of colors among all proper colorings using natural numbers. In this paper, we establish a necessary condition for the existence of graph homomorphisms. Furthermore, we show that \(\Sigma(G) \leq \chi_f(G) |V(G)|\) holds for every graph \(G\).

A connected graph \(G\) is \({k-cycle \; resonant}\) if, for \(0 \leq t \leq k\), any \(t\) disjoint cycles \(C_1, C_2, \ldots, C_t\) in \(G\) imply a perfect matching in \(G – \bigcup_{i=1}^{t} V(C_i)\). \(G\) is \({cycle \; resonant}\) if it is \(k^*\)-cycle resonant, where \(k^*\) is the maximum number of disjoint cycles in \(G\). This paper proves that for outerplane graphs, \(2\)-cycle resonant is equivalent to cycle resonant and establishes a necessary and sufficient condition for an outerplanar graph to be cycle resonant. We also discuss the structure of \(2\)-connected cycle resonant outerplane graphs. Let \(\beta(G)\) denote the number of perfect matchings in \(G\). For any \(2\)-connected cycle resonant outerplane graph \(G\) with \(k\) chords, we show \(k+2 \leq \Phi(G) \leq 2^k + 1\) and provide extremal graphs for these inequalities.

For a bipartite graph the extremal number for the existence of a specific odd (even) length path was determined in J. Graph Theory \(8 (1984), 83-95\). In this article, we conjecture that for a balanced bi-partite graph with partite sets of odd order the extremal number for an even order path guarantees many more paths of differing lengths.The conjecture is proved for a linear portion of the conjectured paths.

Let \(K_{m} – H\) denote the graph obtained from the complete graph on \(m\) vertices, \(K_{m}\), by removing the edge set \(E(H)\) of \(H\), where \(H\) is a subgraph of \(K_{m}\). In this paper, we characterize the potentially \(K_{6} – 3K_{2}\)-graphic sequences, where \(pK_{2}\) is a matching consisting of \(p\) edges.