A note on the extended total graph of commutative rings

Document Type : Research Paper

Authors

¹ University of Tehran

² Department of Mathematics, University of Guilan, Rasht, Iran

Abstract

Let

R

be a commutative ring and

H

a nonempty proper subset of

R

.
In this paper, the extended total graph, denoted by

E T_{H} (R)

is presented, where

H

is a
multiplicative-prime subset of

R

. It is the graph with all elements of

R

as vertices, and for distinct

p, q \in R

, the vertices

p

and

q

are adjacent if and only if

r p + s q \in H

for some

r, s \in R ∖ H

. We also study the two (induced) subgraphs

E T_{H} (H)

and

E T_{H} (R ∖ H)

, with vertices

H

and

R ∖ H

, respectively. Among other things, the diameter and the girth of

E T_{H} (R)

are also studied.

Keywords