The Regular graph of ideals of the commutative ring R, denoted by G reg (R), is a graph whose vertex set is the set of all non-trivial ideals ofR and two distinct vertices I and J are adjacent if and only if either I contains a J -Regular element or J contains an I -Regular element. In this paper, it is proved that the radius of Greg (R) equals 3. The central vertices of G reg (R) are determined, too.

A graph $G$ with a vertex set $V$ and an edge set $E$ is called Regular if the degree of every vertex is the same. A quasi-Regular graph is a graph whose vertices have one of two degrees $r$ and $r-1$, for some positive integer $r$. A graph $G$ is said to be self-complementary if $G$ is isomorphic to it's complement $\overline{G}$. In this paper we give a new method for construction of Regular and quasi-Regular self-complementary graph.

By the Von Neumann Regular graph of R, we mean the graph that its vertices are all elements of R such that there is an edge between vertices x, y if and only if x+y is a von Neumann Regular element of R, denoted by G_Vnr (R). For a commutative ring R with unity, x in R is called Von Neumann Regular if there exists x in R such that a=a2 x. We denote the set of Von Neumann Regular elements by V nr(R). Topological indices are the numbers that is devoted to graphs and show some of their properties. In this paper, first we obtain the degree of vertices for a ring R and the number of edges in different special cases for the ring Z_(p^α ) (p is a prime number) and then we compute Zagreb indices of type one, two and three, Randic, Wiener, Hyper Wiener and reverse Wiener of Von Neumann graph.

A useful tool for investigation various problems in mathematical chemistry and  computational physics is graph entropy.  In this paper, we introduce a new version of graph entropy  and  then we determine it for some classes of graphs.

Let n, t1, …, tk be distinct positive integers. A Toeplitz graph G=(V, E) is a graph with V={1, …, n} and E={(i, j)½½i-j½Î{t1, …, tk}}. In this paper, we present some results on decomposition of Toeplitz graphs.

In this paper we classify distance-Regular graphs, including strongly Regular graphs, ad-mitting a transitive action of the linear groups L(3; 2), L(3; 3), L(3; 4) and L(3; 5) for which the rank of the permutation representation is at most 15. We give details about constructed graphs. In addition, we construct self-orthogonal codes from distance-Regular graphs obtained in this paper.