let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

A graph $G$ is called $P_4$-free, if $G$ does not contain an induced subgraph $P_4$. The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,x)$ is called a domination root of $G$. In this paper we state and prove formula for the domination polynomial of no...

Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph ismaximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximall...

‎In this paper, first we introduce the end of locally finite graphs as an equivalence class of infinite paths in the graph. Then we mention the ends of finitely generated groups using the Cayley graph. It was proved that the number of ends of groups are not depended on the Cayley graph and that the number of ends in the groups is equal to zero, one, two, or infinity. For ...

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_...