In this paper, we consider the sequences \( \{F(n, k)\}_{n \geq k} \) (\(k \geq 1\)) defined by\( F(n, k) = (n – 2)F(n – 1, k) + F(n – 1, k – 1), \quad F(n, 1) = \frac{n!}{2}, \quad F(n, n) = 1. \) We mainly study the log-convexity of \( \{F(n, k)\}{n \geq k} \) (\(k \geq 1\)) when \( k \) is fixed. We prove that \( \{F(n, 3)\}{n \geq 3}, \{F(n, 4)\}{n \geq 5}, \) and \( \{F(n, 5)\}{n \geq 6} \) are log-convex. In addition, we discuss the log-behavior of some sequences related to \( F(n, k) \).
\end{abstract}

Let \( G = C_n \oplus C_n \) with \( n \geq 3 \) and \( S \) be a sequence with elements of \( G \). Let \( \Sigma(S) \subseteq G \) denote the set of group elements which can be expressed as a sum of a nonempty subsequence of \( S \). In this note, we show that if \( S \) contains \( 2n – 3 \) elements of \( G \), then either \( 0 \in \Sigma(S) \) or \( |\Sigma(S)| \geq n^2 – n – 1 \). Moreover, we determine the structures of the sequence \( S \) over \( G \) with length \( |S| = 2n – 3 \) such that \( 0 \notin \Sigma(S) \) and \( |\Sigma(S)| = n^2 – n – 1 \).

Let \(G\) be a finite and simple graph with vertex set \(V(G)\). A nonnegative signed Roman dominating function (NNSRDF) on a graph \(G\) is a function \(f:V(G)\to \{-1,1,2\}\) satisfying the conditions that (i) \(\sum_{x\in N[v]}f(x)\ge 0\) for each \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v\) and (ii)every vertex u for which \(f(u)=-1\) has a neighbor v for which \(f(v)=2\). The weight of an NNSRDF \(f\) is \(\omega(f) = \sum_{v\in V(G)} f(v)\). The nonnegative signed Roman domination number \(\gamma_{sR}^{NN} (G)\) of \(G\) is the minimum weight of an NNSRDF \(G\) In this paper, we initiate the study of the nonnegative signed Roman domination number of a graph and we present different bounds on \(\gamma _{sR}^{NN}(G) \ge (8n-12m)/7\). In addition, if \(G\) is a bipartite graph of order \(n\), then we prove that \(\gamma _{sR}^{NN}(G) \ge^\frac{3}{2}(\sqrt{4n+9}-3)-n\), and we characterize the external graphs.

We consider inverse-conjugate compositions of a positive integer \(n\) in which the parts belong to the residue class of 1 modulo an integer \(m > 0\). It is proved that such compositions exist only for values of \(n\) that belong to the residue class of 1 modulo 2m. An enumerations results is provided using the properties of inverse-conjugate compositions. This work extends recent results for inverse-conjugate compositions with odd parts.

For a graph \(G\) and positive integers \(a_1,…,a_r,\) if every r-coloring of vertics V(G) must result in a monochromatic \(a_1\)-clique of color \(i\) for some \(i \in \{1,…,r\},\) then we write \(G \to (a_1,..a_r)^v\).\(F_v(K_a1,…,K_ar;H)\) is the smallest integer \(n\) such that there is an H-free graph \(G\) of order \(n\), and \(G \to (a_1,…,a_r)^v\). In this paper we study upper and lower bounds for some generalized vertex Folkman numbers of from \(F_v(K_{a1},…,K_{ar};K_4 – e)\), where \(r \in {2,3}\) and \(a_1 \in {2,3}\) for 10 and \(F_v(K_2,K_3;K_4 – e) = 19\) by computing, and prove \(F_v(K_3,K_3;K_4 – e)\ge F_v(K_2,K_2,K_3;K_4 – e)\ge 25\)

We study the complexity of four decision problems dealing with the uniqueness of a solution in a graph: “Uniqueness of a Vertex Cover with bounded size” (U-VC) and “Uniqueness of an Optimal Vertex Cover” (U-OVC), and for any fixed integer \(r \ge 1,\) “Uniqueness of an \(r\)-Dominating Code with bounded size” \((U-DC_r)\) and “Uniqueness of an Optimal \(r\)-Dominating Code” \((U-ODC_r\). In particular, we give a polynomial reduction from “Unique Satisfiability of a Boolean formula” (U-SAT) to U-OVC, and from U-SAT to U-ODC, We prove that U-VC and \(U-DC_r\) have complexity equivalent to that of U-SAT (up to polynomials); consequently, these problems are all \(NP\)-hard, and U-VC and \(U-DC_r\) belong to the class \(DP\).

Let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A signed total Roman dominating function on the digraph \(D\) is a function \(f : V(D)\longrightarrow{-1,1,2}\) \(\sum_{u\in N-(v)} f(u)\ge 1\) for every \(v\in V(D)\), where \(N^{-}(v)\) consists of all inner neighbors of \(v\) for dominating function on \(D\) with the property that \(\sum_{d}^{i=1}f_i(v)\le 1\) for each \(v \in V (D)\) is called a signed total roman dominating family (of functions) on \(D\). The maximum number of functions in a signed total roman dominating family on \(D\)is the signed total Roman domatic number of \(D\). denoted by \(d_{stR}(D)\). In addition, we determine the signed total Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the signed total Roman domatic of graphs.

Let \(G = (V,E)\) be a graph. The transitivity of a graph \(G\), denoted \(Tr(G)\), equals the maximum order \(k\) of a partition \(\pi = \{V_1,V_2,…,V_k\}\) of \(V\) such that for r=every \(i,j,1\le i < j \le k, V_i\) dominates \(V_j\). We consider the transitivity in many special classes of graphs, including cactus graphs, coronas, Cartesian products, and joins. We also consider the effects of vertex or edge deletion and edge addition on the transivity of a graph.

We dedicate this paper to the memory of professor Bohdan Zelinka for his pioneering work on domative of graphs.

The n-dimensional enhanced hypercube \(Q_{n,k}(1 \leq k \leq n-1 )\) is one of the most attractive interconnection networks for parallel and distributed computing system. Let \(H\) be a certain particular connected subgraph of graph \(G\). The \(H\)-structure-connectivity of \(G\), denoted by \(\kappa (G;H),\) is the cardinality of minimal set of subgraphs \(F=\{H_1,H_2,…,H_m\}\) in \(G\) such that every \(H_i\in F\) is isomprphic to \(H\) and \(G-F\) is disconnected. The \(H\)-substructure-connectivity of \(G\), denoted by \(_k^3(G;H)\), is the cardinality of minimal set of subgraphs \(F={H_1,H_2,…,H_m}\) in \(G\) such that every \(H_i\in F\) is isomorphic to a connected subgraph \(H\) , and \(G-F\) is disconnected. Using the structural properties of \(Q_{n,k}\) the \(H\)-structure-connectivity \(\kappa (Q_{n,k};H)\) were determine for \(H \in \{K_1,K_{1,1},K_{1,2},K_{1,3}\}\).

In this paper, we characterize the set of spanning trees of \(G^1_{n,r}\) (a simple connected graph consisting of \(n\) edges, containing exactly one 1-edge-connected chain of \(r\) cycles \(\mathbb{C}^1_r\) and \(G^1_{n,r}\ \mathbb{C}^1_r\) is a forest). We compute the Hilbert series of the face ring \(k[\Delta_s(G^1_{n,r})]\) for the spanning simplicial complex \(\Delta_s (G^1_{n,r})\). Also, we characterize associated primes of the facet ideal \(I_{\mathcal{F}}(\Delta_s(G^1_{n,r})\). Furthermore, we prove that the face ring \(k[\Delta_s(G^1_{n,r})]\) is Cohen-Macaulay.