Spectrum and genus of commuting graphs of some classes of finite rings

We consider commuting graphs of some classes of finite rings and compute their spectrum and genus. We show that the commuting graph of a finite CC-ring is integral. We also characterize some finite rings whose commuting graphs are planar

BIPARTITE GRAPH ASSOCIATED WITH ELEMENTS AND COSETS OF SUBRINGS OF FINITE RINGS

Let  be a finite ring. The bipartite graph associated to elements and cosets of subrings of  is a simple undirected graph  with vertex set , where  is the set of all subrings of , and two vertices  and  are adjacent if and only if  In this study, we investigate some basic properties of the graph . In particular, we investigate some properties of , where  is the ring of matrices over  Also, we study the diameter of the bipartite graph associated to the quaternion rin

Non-Braid Graphs of Ring Zn

The research in graph theory has been widened by combining it with ring. In this paper, we introduce the definition of a non-braid graph of a ring.  The non-braid graph of a ring R, denoted by YR, is a simple graph with a vertex set R\B(R), where B(R) is the set of x in R such that  xyx=yxy for all y in R.  Two distinct vertices x and y are adjacent if and only if xyx not equal to yxy.  The method that we use to observe the non-braid graphs of Zn is by seeing the adjacency of the vertices and its braider.  The main objective of this paper is to prove the completeness and connectedness of the non-braid graph of ring Zn. We prove that if n is a prime number, the non-braid graph of Zn is a complete graph. For all n greater than equal to 3,  the non-braid graph of Zn is a connected graph

Some properties of compatible action graph

In this paper, the compatible action graph for the finite cyclic groups of p-power order has been considered. The purpose of this study is to introduce some properties of the compatible action graph for finite p-groups

On r-Noncommuting Graph of Finite Rings

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if x,y≠r and x,y≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6