let g=(v,e) be a simple graph. an edge labeling f:e to {0,1} induces a vertex labeling f^+:v to z_2 defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$, where z_2={0,1} is the additive group of order 2. for $iin{0,1}$, let e_f(i)=|f^{-1}(i)| and v_f(i)=|(f^+)^{-1}(i)|. a labeling f is called edge-friendly if $|e_f(1)-e_f(0)|le 1$. i_f(g)=v_f(1)-v_f(0) is called the edge-f...

Let $G$ be a finite group and $cd^*(G)$ be the set of nonlinear irreducible character degrees of  $G$. Suppose that $rho(G)$ denotes the set of primes dividing some element of $cd^*(G)$. The bipartite divisor graph for the set of character degrees which is denoted by $B(G)$, is a bipartite graph whose vertices are the disjoint union of $rho(G)$ and $cd^*(G)$, and a vertex $p in rho(G)$ is conne...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

a finite group $g$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. taeri conjectured that an insoluble group satisfying this condition is isomorphic to $s times a$ where $a$ is abelian and $s cong psl_2(q)$ for $q in {4,8}$. we confirm this conjecture.

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...

for a graph g, let d(g) be the set of all strong orientations of g. the orientationnumber of g is min {d(d) |d belongs to d(g)}, where d(d) denotes the diameterof the digraph d. in this paper, we determine the orientation number for some subgraphs of complete bipartite graphs.

given a graph $g$, let $g^sigma$ be an oriented graph of $g$ with the orientation $sigma$ and skew-adjacency matrix $s(g^sigma)$. then the spectrum of $s(g^sigma)$ consisting of all the eigenvalues of $s(g^sigma)$ is called the skew-spectrum of $g^sigma$, denoted by $sp(g^sigma)$. the skew energy of the oriented graph $g^sigma$, denoted by $mathcal{e}_s(g^sigma)$, is defined as the sum of the n...

Abstract Graphs have been widely adopted in various fields, where many graph models are developed. Most of previous research focuses on unipartite or homogeneous analysis. In this graphs, the relationships between same type entities preserved graphs. Meanwhile, bipartite graphs that model complex among different with vertices partitioned into two disjoint sets, becoming increasing popular and u...