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.

The distance $d(u,v)$ between vertices $u$ and $v$ of a connected graph $G$ is equal to the number of edges in a minimal path connecting them. The transmission of a vertex $v$ is defined by $\sigma(v)=\sum\limits_{u\in V(G)}{d(v,u)}$. A topological index is said to be a transmission-based topological index (TT index) if it includes the transmissions $\sigma(u)$ of vertices of $G$. Because $\sigma(u)$ can be derived from the distance matrix of $G$, it follows that transmission-based topological indices form a subset of distance-based topological indices.In this article we survey some results on the computation of some transmission-based graph invariants of intersection graph, hypercube graph, Kneser graph, unitary Cayley graph and Paley graph.

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.

An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for ij, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. Balaban introduced some monster graphs and then Randic computed complexity indices of them (1973, 2001). In this paper, with a simple method, we calculate the Automorphism group of some weighted graphs.

The aim of this paper is to determine an upper bound for the number of non-co-spectral permutation graphs in terms of Automorphism group of a graph G. As a corollary, we determine the eigenvalues of all permutation graphs P (Cn), where  2 Aut(Cn).

The coprime graph  GG with a finite group G as follows: Take G as the vertex set of GG and join two distinct vertices u and v if (|u|, |v|) =1. In the paper, we explore how the graph theoretical properties of GG can effect on the group theoretical properties of G.