This paper studies a repetitive polling game played on an n-vertex graph G. At first, each vertex is colored, Black or White. At each round, each vertex (simultaneously) recolors itself by the color of the majority of its closed neighborhood. The variants of the model differ in the choice of a particular tiebreaking rule. We assume the tie-breaking rule is Prefer-White and we study the relation between the notion of “dynamic monopoly” and “vertex cover” of G. In particular, we show that any vertex cover of G is a dynamic monopoly or reaches a 2−periodic coloring. Moreover, we compute dyn(G) for some special classes of graphs including paths, cycles and links of some graphs.

The purpose of this paper is the study of non-weak commutative hypergroups associated with hypergraphs. In this regards, we construct a hyperoperation on the set of vertices of hypergraph and obtain some results and characterizations of them. Moreover, according to this hyperoperation, we investigate conditions under which the hypergroupoid is a join space hypergroup. Finally, we present an application to marketing social network.

In this paper, an algorithm for edge-coloring of Simple graphs has been discussed. Simplicity and quality of this algorithm compare to other existing algorithms and the privileges are shown.

The numerical range of a Simple graph G, named F(G), is the numerical range of its adjacency matrix A(G). The main purpose of this paper is to approximate F(G). Then, using this approximation, bounds for the largest and the smallest eigenvalues of G are proposed. In fact, lower bounds for the largest eigenvalues of G are presented in terms of disjoint induced subgraphs of G and the numerical range of the square of A(G).

The first (P1) and the second (P2) multiplicative Zagreb indices of a connected graph G, with vertex set V(G) and edge set E(G), are defined as P1(G)=PuÎV(G) du2 and P2(G)=PuvÎE(G) dudv, respectively, where du denotes the degree of the vertex u. In this paper we present a Simple approach to order these indices for connected graphs on the same number of vertices. Moreover, as an application of this Simple approach, we extend the known ordering of the first and the second multiplicative Zagreb indices for some classes of connected graphs.

For a coloring c of a graph G, the edge-di erence coloring sum and edge-sum coloring sum with respect to the coloring c are respectively Σ c D(G) = Σ jc(a) 􀀀 c(b)j and Σ s S(G) = Σ (c(a) + c(b)), where the summations are taken over all edges ab 2 E(G). The edge-di erence chromatic sum, denoted by Σ D(G), and the edge-sum chromatic sum, denoted by Σ S(G), are respectively the minimum possible values of Σ c D(G) and Σ c S(G), where the minimums are taken over all proper coloring of c. In this work, we study the edge-di erence chromatic sum and the edge-sum chromatic sum of graphs. In this regard, we present some necessary conditions for the existence of homomorphism between two graphs. Moreover, some upper and lower bounds for these parameters in terms of the fractional chromatic number are introduced as well.

The problem of path detection in graphs has been proposed from the past up to present, and various solutions have been proposed for this purpose, but it is often not an easy task to implement these methods on a computer. In this paper, a technique for detecting paths in a graph will be introduced using matrix algebra, which makes it possible to implement this rule on a computer. This method can be helpful the optimization of tree-spanning trees in networks. At the end of this study, a numerical example is solved using the proposed method.

‎In this article‎, ‎we employ a novel and unique method to analyze the Eulerian nature of the power graphs $ P_{i}(D) $ for 3 ⩽ i ⩽ 6‎. ‎Then‎, ‎we will mention some applications of Eulerian power graphs in computer networks‎.

Let G be a , nite group and cd(G) denote the character degree set for G. The prime graph ∆, (G) is a Simple graph whose vertex set consists of prime divisors of elements in cd(G), denoted , (G). Two primes p,q 2 , (G) are adjacent in ∆, (G) if and only if pq j a for some a 2 cd(G). We determine which Simple 4-regular graphs occur as prime graphs for some , nite nonsolvable group.