Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...

An n-vertex graph is prime if it has no homogeneous set, that is a set X of vertices (2 ≤ |X| ≤ n − 1) such that every vertex not in X is either complete or anticomplete to X. A chain of length t is a sequence of t+ 1 vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors...

It has long been known that the class of connected nonbipartite graphs (with loops allowed) obeys unique prime factorization over the direct product of graphs. Moreover, it is known that prime factorization is not necessarily unique in the class of connected bipartite graphs. But any prime factorization of a connected bipartite graph has exactly one bipartite factor. It has become folklore in s...

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of #pHomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating t...

In this last of three papers on simplicial tree-decompositions of graphs we investigate the extent to which prime factors in such decompositions are unique, or depend on the decomposition chosen. A simple example shows that a prime decomposition of a graph may have superfluous factors, the omission of which leaves a set of factors that can be rearranged into another decomposition of the same gr...

AbstractThe notion of the prime graph a ring R was first introduced by Bhavanari, Kuncham, and Dasari in 2010. The R, denoted PG(R) is whose vertices are all elements ring, where two distinct x y adjacent if only xRy = 0 or yRx 0. In this paper, we study forms properties integer modulo group, some examples number its spanning trees. it found that for n, maximum degree PG(Z_n) n-1 minimum 1. The...

The prime index graph π(G) of a finite group G is special type undirected simple whose vertex set subgroups G, in which two distinct vertices are adjacent if one has the other. Let p and q be primes. In this paper, we establish that cyclic p-group Zpn, abelian Zpn × Zq Zp always have graceful labeling without any condition on n using concept path or p-layer ladder size + 1.

A regular graph is called semisymmetric if it is edge-transitive but not vertex-transitive. It is proved that the Gray graph is the only cubic semisymmetric graph of order 2p3, where p 3 is a prime.

A regular graph is called semisymmetric if it is edge transitive but not vertex transitive It is proved that the Gray graph is the only cubic semisymmetric graph of order p where p is a prime