** Authors:**
ALI RAMIN

** Abstract: **
Let R be a commutative ring with Z(R), its set of
zero-divisors and \mbox{Reg}(R), its set of regular elements. Total
graph of R, denoted by T(\Gamma(R)), is the graph with all elements of
R as vertices, and two distinct vertices x,y \in R, are adjacent in
T(\Gamma(R)) if and only if x+y \in Z(R). In this paper, some
properties of T(\Gamma(R)) have been investigated, where R is a finite
commutative ring and a new upper bound for vertex-connectivity has
been obtained in this case. Also, we have proved that the
edge-connectivity of T(\Gamma(R)) coincides with the minimum degree if
and only if R is a finite commutative ring such that Z(R) is not an
ideal in R.

** Keywords: **
Commutative rings, total graph, regular elements, zero-divisors

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