Let G = (V,E) be a simple graph with exactly n vertices and m edges. The aim of this paper is a new method for investigating nontriviality of the Automorphism group of graphs. To do this, we prove that if |E| ³ ë (n - 1)2/2û then |Aut(G)| > 1 and |Aut(G)| is even number.

In the present paper, we introduce the Automorphism group of cubic polyhedral graphs whose faces are triangles, quadrangles, pentagons and hexagons.

Let G be a group and R,L,S be subsets of G such that $R=R^{-1}$, $L=L^{-1}$ and $1\notin R\cup L$. The undirected graph $\SC(G;R,L,S)$ with vertex set  union of $G_1=\{g_1\mid g\in G\}$ and $G_2=\{g_2\mid g\in G\}$, and edge set the union of $\{\{g_1,(gr)_1\}\mid g\in G, r\in R\}$, $\{\{g_2,(gl)_2\}\mid g\in G,l\in L\}$ and $\{\{g_1,(gs)_2\}\mid g\in G,s\in S\}$ is called semi-Cayley graph over G.  We say that $\SC(G;R,L,S)$ is quasi-abelian if R,L and S are a  union of conjugacy classes of G. In this paper, we study the Automorphism group of quasi-abelian semi-Cayley graphs.

Let $L$ be a lattice and $S$ be a $\wedge$-closed subset of $L$. The graph $\Gamma_{S}(L)$ is a simple graph with all elements of $L$ as vertex set and two distinct vertex $x,y$ are adjacent if and only if $x\vee y\in S$. In this paper, we verify the Automorphism group of $\Gamma_{S}(L)$ and the relation by Automorphism group of the lattice $L$. Also we study some properties of the graph $\Gamma_{S}(L)$ where $S$ is a prime filter or an ideal such as the perfect maching.

Let F be an (α, β)-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group G. We consider the Automorphism and isometry groups of the Finsler manifold (G, F) and their intersection. We prove that for an arbitrary left invariant vector field X and any compact subgroup K of Automorphisms which X is invariant under them, there exists an (α, β)-metric such that K is a subgroup of its isometry group.