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.

We establish formulas for the number of all downsets (or equivalently, of all antichains) of a finite poset \(P\). Then, using these numbers, we determine recursively and explicitly the number of all posets having a fixed set of minimal points and inducing the poset \(P\) on the non-minimal points. It turns out that these counting functions are closely related to a collection of downset numbers of certain subposets. Since any function ar that kind is an exponential sum (with the number of minimal points as exponents), we call it the exponential function of the poset. Some linear equalities, divisibility relations, upper and lower bounds. A list of all such exponential functions for posets with up to five points concludes the paper.

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The first Zagreb index of a graph is defined as the sum of squares of the degrees of the vertices of the graph. The second Zagreb index of a graph is defined as the sum of products of the degrees of a pairs of the adjacent vertices of the graph. In this paper, we establish some sufficient conditions for a nearly balanced bipartite graph with large minimum degree to be traceable in terms of the energy, the first Zagreb index and the second Zagreb index of the quasi-complement of the graph, respectively.

In \(PG(3,P^{2^h}),\) ovoids and cones projecting an oval from a point are characterized as three character sets with respect to lines and planes, respectively.

There are 19 connected cubic graphs of order 10. If \(G\) is one of a specific set 3 the 19 graphs. we find necessary and sufficient conditions for the existence of \(G\)-decompositions of \(K_v\).

For a positive integer \(k,\) let \( [k] = {1,2,…,k}\), let \(P([k])\) denote the power set of the set \([k]\) and let \(P*([k]) = P([k]) – {\emptyset}\). For each integer \(t\) with \(1 \le t < k\), let \(P_t([k])\) denote the set of \(t\)-element subsets of \(P([k])\). For an edge coloring \(c : E(G)\to P_t ([k])\) of a graph \(G\), where adjacent edges may be colored the same, \(c' : V(G) \to P*([k])\) is the vertex coloring in which \(c' (v)\) is the union of the color sets of the edges incident with \(v\). If \(c'\) is a proper vertex coloring of \(G\), then \(c\) is a majestic \(t\)-tone k-coloring of \(G\) For a fixed positive integer \(t\), the minimum positive integer \(k\) for which a graph \(G\) has a majestic t-tone k-coloring is the majestic t-tone index \(maj_t (G)\) of \(G\). It is known that if \(G\) is a connected bipartite graph or order at least 3, then \(maj_t(G) = t + 1\) or \(maj_t (G) = t + 2\) for each positive integer t. It is shown that (i) if \(G\) is a 2-connected bipartite graph of arbitrarily large order \(n\) whose longest cycles have length \(l\) where where \(n-5 \leq l \leq n\) and \(t\geq 2\) is an integer, then \(maj_t(G)=t+1\) and (ii) there is a 2-connected bipartite graph F of arbitrarily large order n whose longest cycles have length n-6 and \(maj_2(F)=4\). Furthermore, it is shown for integers \(k,t \ge 2\) that there exists a k-connected bipartite graph \(G\) such that \(maj_t(G) =t+2\). Other results and open questions are also presented.

The 3-path \(P_3(G)\) of a connected graph \(G\) of order 3 or more has the set of all 3-path (path of order 3) of \(G\) as its vertex of \(P_3(G)\) are adjacent if they have a 2-path in common. A Hamiltonian walk in a nontrivial connected graph \(G\) is a closed walk of minimum length that contains every vertex of \(G\). With the aid of spanning trees and Hamiltonian walks in graphs, we provide sufficient conditions for the 3-path graph of a connected graph to be Hamiltonian.

Let \(R\) be a commutative ring with identity. For any integer \(K > 1,\) an element is a \(k\) zero divisor if there are \(K\) distinct elements including the given one, such that the product of all is zero but the product of fewer than all is nonzero. Let \(Z(R,K)\) denote the set of the \(K\) zero divisors of \(R\). A ring with no \(K\)-zero divisors is called a \(K\)-domain. In this paper we define the hyper-graphic constant \(HG(R)\) and study some basic properties of \(K\)-domains. Our main results is theorem 5.1 which is as fellow:

Let \(R\) be a commutative ring such that the total ring of fraction \(T(R)\) is dimensional. If \(R\) is a \(K\)-domain for \(k \geq 2,\) then \(R\) has finitely many minimal prime ideals.

Using the results and lemma 5.4, we improve a finiteness theorem proved in [11] to a more robust theorem 5.5 which says:

Suppose \(R\) is not a \(k\)-domain and has more then \(k\)-minimal prime ideals.

Further, suppose that \(T(R)\) is a zero dimensional ring. Then \(Z(R,K)\) is finite if and only if \(R\) is finite.

We end this paper with a proof of an algorithm describing the maximal \(k\)-zero divisor hypergraphs on \(\mathbb{Z}_n\).