INVERSE SIGNED DOMINATING FUNCTIONS OF CORONA AND ROOTED PRODUCT GRAPHS

: Graph theory is an interesting subject in mathematics. Applications in many fields like Linguistics, Engineering communications, Physical Sciences, Coding theory, Computer networking and Logical Algebra. The theory of domination in graphs has a wide range of applications. Among these applications, the most often discussed is a coding theory and communication networks. Inverse domination theory of graphs which are the important branches of graph theory. In this paper, we study the maximal inverse signed dominating functions of corona product graph of a path with a complete graph and rooted product graph of a path with a cycle.


INTRODUCTION
Mostly Product of graphs occurs in discrete mathematics. In 1970, Frucht & Harary [6] introduced a new product on two graphs G 1 and G 2 , called corona product denoted by 1 2 G G . The corona product of a path n P with a complete graph m K is a graph obtained by taking one copy of nvertex path n P and n copies of m K and then joining the th i vertex of n P to every vertex of th i copy of m K and it is denoted by n m P K , where n>0 and m>0 . In 1978, Godsil and McKay [1] introduced a new product on two graphs 1 G and 2 G , called rooted product denoted by 1 2 G G  . In this paper we consider the rooted product graph like, here n P be a Path graph with n vertices and ( 3) m C m be a cycle with a sequence of n rooted graphs 1  contains (m-1) vertices in each copy of m C . In 1995, Dunbar, Hedetniemi, Henning and Slater [4] have studied about "Signed Domination in Graphs". Further we studied about signed domination in [2,7]. In 1996, Favaron [5] have studied about "Signed domination in regular graphs". In 2010, Zhong-sheng [3] have studied about "On Inverse Signed Total Domination in Graphs". By using signed domination related parameters we can find out inverse signed domination parameters on product graphs. Here +1 is assigned to

RESULTS ON ROOTED PRODUCT GRAPH
vertices in each copy of m C in G, -1 is assigned to all other vertices in G. Case 1: Suppose n v P  be such that If ( ) (ii) Then   N v contains 3 vertices of m C and zero vertex of n P in G .
From the above cases the function f is an inverse signed Here two cases are followed.
From the above cases, we get This implies that the function g is not an inverse signed dominating function. Hence f is a maximal inverse signed dominating function on G. Now inverse signed total domination number is the sum of the function value of all vertices in G, that is vertices and | | E number of edges.

Case I: Suppose m=3k+1
Let : vertices in each copy of m C in G, -1 is assigned to all other vertices in G. Case 1: Suppose n v P  be such that (i) Here   N v contains 2 vertices of m C and one vertex of n P in G .
(ii) Here   N v contains 3 vertices of m C and zero vertex of n P in G .
Here two cases are followed. Case 3: Suppose n v P  be such that (ii)Here   N v contains 3 vertices of m C and zero vertex of From the above cases, we get [ ] ( ) 0,forsome .
This implies that the function g is not an inverse signed dominating function. Hence f is a maximal inverse signed dominating function on G. Now inverse signed total domination number is the sum of the function value of all vertices in G, that is

Case II: Suppose m=3k+2
Let : (i) Here   N v contains 2 vertices of m C and one vertex of (ii) Here   N v contains 3 vertices of m C and zero vertex of n P in G then ( )