Let $G$ be a finite group and $pi_{e}(G)$ be the set of orders of all elements in $G$. The set $pi_{e}(G)$ determines the prime graph (or Grunberg-Kegel graph) $Gamma(G)$ whose vertex set is $pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$ and $q$ are adjacent if and only if $pqinpi_{e}(G)$. The degree $deg(p)$ of a vertex $pin pi(G)$, is the number of edges incident...

Let $G$ be a finite group and $pi(G)$ be the set of all the prime‎ ‎divisors of $|G|$‎. ‎The prime graph of $G$ is a simple graph‎ ‎$Gamma(G)$ whose vertex set is $pi(G)$ and two distinct vertices‎ ‎$p$ and $q$ are joined by an edge if and only if $G$ has an‎ ‎element of order $pq$‎, ‎and in this case we will write $psim q$‎. ‎The degree of $p$ is the number of vertices adjacent to $p$ and is‎ ...

let $g$ be a finite group and $pi(g)$ be the set of all the prime‎ ‎divisors of $|g|$‎. ‎the prime graph of $g$ is a simple graph‎ ‎$gamma(g)$ whose vertex set is $pi(g)$ and two distinct vertices‎ ‎$p$ and $q$ are joined by an edge if and only if $g$ has an‎ ‎element of order $pq$‎, ‎and in this case we will write $psim q$‎. ‎the degree of $p$ is the number of vertices adjacent to $p$ and is‎ ...

if $g$ is a connected graph with vertex set $v$, then the eccentric connectivity index of $g$, $xi^c(g)$, is defined as $sum_{vin v(g)}deg(v)ecc(v)$ where $deg(v)$ is the degree of a vertex $v$ and $ecc(v)$ is its eccentricity. in this paper we show some convergence in probability and an asymptotic normality based on this index in random bucket recursive trees.

the first zagreb index, $m_1(g)$, and second zagreb index, $m_2(g)$, of the graph $g$ is defined as $m_{1}(g)=sum_{vin v(g)}d^{2}(v)$ and $m_{2}(g)=sum_{e=uvin e(g)}d(u)d(v),$ where $d(u)$ denotes the degree of vertex $u$. in this paper, the firstand second maximum values of the first and second zagreb indicesin the class of all $n-$vertex tetracyclic graphs are presented.

The geometric-arithmetic index is another topological index was defined as 2 deg ( )deg ( ) ( ) deg ( ) deg ( ) G G uv E G G u v GA G u v     , in which degree of vertex u denoted by degG (u). We now define a new version of GA index as 4 ( ) 2 ε ( )ε ( ) ( ) ε ( ) ε ( ) G G e uv E G G G u v GA G   u v    , where εG(u) is the eccentricity of vertex u. In this paper we compute this new t...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

The Wiener index W(G) of a connected graph G is defined as the sum of the distances between all unordered pairs of vertices of G. The eccentricity of a vertex v in G is the distance to a vertex farthest from v. In this paper we obtain the Wiener index of a graph in terms of eccentricities. Further we extend these results to the self-centered graphs.

Let G be a simple connected graph with the vertex set V = V(G) and the edge set E = E(G), without loops and multiple edges. For counting qoc strips in G, Omega polynomial was introduced by Diudea and was defined as Ω(G,x ) = ( , ). , c c m G c x  where m(G,c) be the number of qoc strips of length c in the graph G. Following Omega polynomial, the Sadhana polynomial was defined by Ashrafi et al ...