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Abstract

The well-known problems of graph theory are minimum covering energy  and minimum dominating energy . A total dominating set of a graph , is a subset S of V such that each vertex V-S is adjacent to at least one vertex of S. Let the minimum total dominating set of the graph H be S. Then  is called inverse total dominating set of H, if  contains a total dominating set  of H. The minimum number of vertices in inverse total dominating set of graph H is called an inverse total domination number . The sum of absolute values of the eigen values of adjacency matrix is defined as the energy of the graph. In this journal, we introduced Minimum Inverse Total dominating energy of a graph and also, we computed minimum inverse total dominating energy of a Cycle Graph, Star Graph, Wheel Graph and Complete graph. The Minimum inverse total dominating energy is also defined for the special classes of graph such as Chvatal graph, Octahedron graph and Paley Graph.

Keywords: minimum inverse total dominating set, dominating set, minimum inverse total dominating matrix, minimum inverse total dominating energy, minimum inverse total dominating eigen values.

1 Introduction

The inspiration of description energy of graph happened from quantum Chemistry. During 1930s, E. Hückel presented chemical applications of graph theory in his molecular orbital theory where eigenvalues of graphs take place. In quantum chemistry, the skeleton of non-saturated hydrocarbon is represented by a graph. The energy levels of electrons in such a molecule are eigenvalues of graph. The carbon atoms and chemical bond between them in a hydrocarbon system denote vertices and edges, respectively, in a molecular graph. A lot of work has been done on graph theory, special graph labeling, chemical graph theory and graph energies

The total  electron energy of conjugated hydrocarbon molecules is closely connected to the graph invariant. The Energy of graph was introduced by Gutman for a simple graph. In the beginning only very, few mathematicians show their interest in this concept. Later, energy of graph becomes one of the interesting topics.

Let H be a graph and the adjacency matrix of a graph be . Let the eigenvalues of adjacency matrix of a graph H be  . The sum of absolute values of eigenvalues of graph H is defined as the energy  of graph, ie,  [10][11][12][13]

A total dominating set of a graph , is a subset S of V such that each vertex V-S is adjacent to at least one vertex of S. Let the minimum total dominating set of H be S. Then  is called inverse total dominating set of H, if  contains a total dominating set  of H. The minimum number of vertices in inverse total dominating set of H is called as inverse total domination number .

In this journal, we introduced Minimum Inverse Total dominating energy of a graph and also, we computed minimum inverse total dominating energy of a Cycle Graph, Star Graph, Wheel Graph and Complete graph. The Minimum inverse total dominating energy is also defined for the special classes of graph such as Chvatal graph, Octahedron graph and Paley Graph.