Author/Authors :
Esmaeili Khalil Saraei, F Fouman Faculty of Engineering - College of Engineering - University of Tehran , Navidinia, E. Department of Mathematics - Faculty of Mathematical Sciences - University of Guilan
Abstract :
Let
M
M be a module over a commutative ring
R
R and
U
U a nonempty proper subset of
M
M.
In this paper, the extended total graph, denoted by
E
T
U
(
M
)
ET
U
(M), is presented, where
U
U is a
multiplicative-prime subset of
M
M. It is the graph with all elements of
M
M as vertices, and for distinct
m
,
n
∈
M
m,n∈M, the vertices
m
m and
n
n are adjacent if and only if
r
m
+
s
n
∈
U
rm+sn∈U for some
r
,
s
∈
R
∖
(
U
:
M
)
r,s∈R∖(U:M). We also study the two (induced) subgraphs
E
T
U
(
U
)
ET
U
(U) and
E
T
U
(
M
∖
U
)
ET
U
(M∖U), with vertices
U
U and
M
∖
U
M∖U, respectively. Among other things, the diameter and the girth of
E
T
U
(
M
)
ET
U
(M) are also studied.
Keywords
