the zero-divisor graph of a commutative ring r with respect to nilpotent elements is a simple undirected graph $gamma_n^*(r)$ with vertex set z_n(r)*, and two vertices x and y are adjacent if and only if xy is nilpotent and xy is nonzero, where z_n(r)={x in r: xy is nilpotent, for some y in r^*}. in this paper, we investigate the basic properties of $gamma_n^*(r)$. we discuss when it will be eu...

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

Let $R$ be a commutative ring with identity, and $ mathrm{A}(R) $ be the set of ideals with non-zero annihilator. The annihilating-ideal graph of $ R $ is defined as the graph $AG(R)$ with the vertex set $ mathrm{A}(R)^{*}=mathrm{A}(R)setminuslbrace 0rbrace $ and two distinct vertices $ I $ and $ J $ are adjacent if and only if $ IJ=0 $. In this paper, conditions under which $AG(R)$ is either E...

Labeled graphs, in which edges are labeled by letters from some alphabet Σ, are extensively used to model many types of relations associated with actions, costs, owners, or other properties. Each path in a labeled graph induces a word in Σ∗ – the one obtained by concatenating the letters along the edges in the path. Classical graph-theory problems give rise to new problems that take these words...

A perfect path double cover (PPDC) of a graph G is a family P of paths of G such that every edge of G belongs to exactly two paths of P and each vertex of G occurs exactly twice as an endpoint of a path in P. Li (J. Graph Theory 14 (1990) 645–650) has shown that every simple graph has a PPDC.A regular perfect path double cover (RPPDC) of a graph G is a PPDC of G in which all paths are of the sa...

in the process of structural analysis we often come to structures that can be analyzed with simpler methods than the standard approaches. for these structures, known as regular structures, the matrices involved are in canonical forms and their eigen-solution can be performed in a simple manner. however, by adding or removing some elements or nodes, such methods cannot be utilized. here, an effi...

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

We characterize the graphs that admit a decomposition into circuits, i.e. finite or infinite connected 2-regular graphs. Moreover, we show that, as is the case for the removal of a closed eulerian subgraph from a finite graph, removal of a non-dominated eulerian subgraph from a (finite or infinite) graph does not change its circuit-decomposability or circuit-indecomposability. For cycle-decompo...

A graph is subeulerian if it is spanned by an eulerian supergraph. Boesch, Suffel and Tindell have characterized the class of subeulerian graphs and determined the minimum number of additional lines required to make a subeulerian graph eulerian. In this paper, we consider the related notion of a subsemi-eulerian graph, i.e. a graph which is spanned by a supergraph having an open trail containin...