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Let be a simple graph with vertex set V and edge set E. A function is said to be a reverse total signed vertex dominating function if for every , the sum of function values over v and the elements incident to v is less than zero. In this paper, we present some upper bounds of reverse total signed vertex domination number of a graph and the exact values of reverse total signed vertex domination number of circles, paths and stars are given .

In this paper we shall use the terminology of [

.

Many domination parameters in graphs has been studied richly [2-4] A function is a signed dominating function if for every vertex

,.

The weight of is the sum of the function values of all vertices in. The signed domination number of is the minimum weight of signed dominating functions on. This concept was introduced by Dunbar et al. [

Definition 1. Let be a simple graph. A reverse total signed vertex dominating function of is a function such that

for all. The reverse total signed vertex domination number of, denoted by, is the maximum weight of a reverse total signed vertex dominating function of. A reverse total signed vertex dominating function is called a -function of if .

Proposition 1 For any graph,

.

Proof. Let be a -function of. Then

.

Let

,

,

,

.

Then

.

Therefore.

Propositon 2 For any graph,.

Proof. Let be a -function of. Then for every, and we have

Thus.

Propositon 3 For any graph,.

Proof. Let be a -function of., , and are defined as Proposition 2. Then

.

We define two induced graphs and of as follows:

, ,.

Then for every,

and. For every, we have

and. Thus

Therefore

Since

we have. Therefore.

Propositon 4 For any star,.

Proof. Let be a -function. Let

,

,

where is the center of. Since for every, , we have

.

On the other hand, consider the function

such that

,.

Then is a reverse total signed vertex dominating function on and

.

Thus, which implies that .

Propositon 5 For any circle,.

Proof. Let be a -function of. Let

,.

Since for every, , we have

.

Thus

.

Therefore.

On the other hand, consider the mapping

such that

,.

Then is a reverse total signed vertex dominating function on and. Therefore

which implies.

Propositon 6 For any complete bipartite graph,.

Proof. Letbe a -function. Let

, ,

and

.

Since for every, , we have . Therefore

.

On the other hand, consider the mapping

such that, for,

for and. Then is a reverse total signed vertex dominating function on

and. Thereforewhich implies.

This work was supported by the Natural Science Foundation of Hebei Province (A2012408002), the Educational Commission of Hebei Province (ZH2011122, Z2011157) and Langfang Teachers College (LSZQ201106).