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.

Let D be the diameter and dG(vi, vj ) be the distance between the vertices vi and vj of a connected graph G. The complementary distance matrix of a graph G is CD(G) = [cdij ] in which cdij = 1 +D − dG(vi, vj ) if i 6= j and cdij = 0 if i = j. The complementary transmission CTG(v) of a vertex v is defined as CTG(v) = Pu2V (G)[1+D− dG(u, v)]. Let CT(G) = diag[CTG(v1), CTG(v2), . . ., CTG(vn)]. The complementary distance signless Laplacian matrix of G is CDL+(G) = CT(G) + CD(G). In this paper, we obtain the bounds for the largest eigenvalue of CDL+(G). Further we determine Nordhaus-Gaddum type results for the largest eigenvalue. We also establish some bounds for the complementary distance signless Laplacian energy.

The eccentricity of a vertex v of G is the largest distance between v and any other vertex in G. The reciprocal complementary Wiener (RCW) index of G is defined as..

We use the recursive method of construction large sets of t-designs given by Qiu-rong Wu (A note on extending t-designs, Australas. J. Combin., 4 (1991) 229 {235.), and present a similar method for constructingt -subset-regular self-complementary k -uniform hypergraphs of order v. As an application we show the existence of a new family of 2-subset-regular self-complementary 4-uniform hypergraphs with u=16 m+3.

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.

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.

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.