On r-Noncommuting Graph of Finite Rings

Abstract

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.

Quantum Circuit Template Matching Optimization Method for Constrained Connectivity

“…As from the reference [27], the linear substructure of a quantum topology can be expressed as an r-noncommutative graph on finite rings. Therefore, Algorithm 3 is suited for determining the optimal path for any given quantum topology that is represented by r-noncommutative graph on finite rings.…”

Relative g-noncommuting graph of finite groups

“…The ring theoretic analogues of Γ X,G , Γ g G and Γ g X,G can be found in [6,17] and [22] respectively. Let G 1 + G 2 be the join of the graphs G 1 and G 2 and let G be the complement of G. Then we have the following observations, where K n is the complete graph on n vertices and deg(v) denotes the degree of any vertex v in Γ g X,G .…”

On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups

“…Characterizations of finite groups through various graphs defined on them have been an interesting topic of research over the last five decades. The non-commuting graph is one of such interesting graphs widely studied in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] since its inception [17]. In this paper, we introduce a generalization of non-commuting graph of a finite group.…”