let $g$ be a simple graph with an orientation $sigma$‎, ‎which ‎assigns to each edge a direction so that $g^sigma$ becomes a‎ ‎directed graph‎. ‎$g$ is said to be the underlying graph of the‎ ‎directed graph $g^sigma$‎. ‎in this paper‎, ‎we define a weighted skew‎ ‎adjacency matrix with rand'c weight‎, ‎the skew randi'c matrix ${bf‎ ‎r_s}(g^sigma)$‎, ‎of $g^sigma$ as the real skew symmetric mat...

the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. t...

Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

Let G^s be a signed graph, where G = (V;E) is the underlying simple graph and s : E(G) to {+, -} is the sign function on E(G). In this paper, we obtain k-th signed spectral moment and k-th signed Laplacian spectral moment of Gs together with coefficients of their signed characteristic polynomial and signed Laplacian characteristic polynomial are calculated.

Let G be a graph. The first Zagreb polynomial M1(G, x) and the third Zagreb polynomial M3(G, x) of the graph G are defined as:     ( ) ( , ) [ ] e uv E G G x x d(u) + d(v) M1 , ( , )  euvE(G) G x x|d(u) - d(v)| M3 . In this paper, we compute the first and third Zagreb polynomials of Cartesian product of two graphs and a type of dendrimers.

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...

let $g$ be a finite group. we construct the prime graph of $ g $,which is denoted by $ gamma(g) $ as follows: the vertex set of thisgraph is the prime divisors of $ |g| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ g $ contains anelement of order $ pq $.in this paper, we determine finite groups $ g $ with $ gamma(g) =gamma(l_3(q)) $, $2 leq q < 100 $ and prov...

Conceptually graph vulnerability relates to the study of graphintactness when some of its elements are removed. The motivation forstudying vulnerability measures is derived from design and analysisof networks under hostile environment. Graph tenacity has been anactive area of research since the the concept was introduced in1992. The tenacity T(G) of a graph G is defined asbegin{center} $T(G)=di...

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....