Independent Transversal Dominating Energy of a Graph
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Let G be a simple graph and let V (G) be its vertex set. The independent transversal dominating set or γit-set is a dominating set of G which intersects every maximal independent set of G. Using the γit-set, a diagonal matrix is constructed which is similar to the ones available in the literature for covering set, equitable dominating set and so on. Using the idea of signless Laplacian, one can add this diagonal matrix to the usual adjacency matrix of G to obtain new matrix called independent transversal dominating adjacency matrix denoted by Ait(G). The sum of all the absolute values of eigenvalues of Ait(G) is called independent transversal dominating energy denoted by Eit(G). In the present paper, some spectral properties of Ait(G) are obtained. Some upper and lower bounds for the largest eigenvalue of Ait(G) and Eit(G) are derived. Eit(G) is obtained for some standard graphs.
Abstract
Let G be a simple graph and let V (G) be its vertex set. The independent transversal dominating set or γit-set is a dominating set of G which intersects every maximal independent set of G. Using the γit-set, a diagonal matrix is constructed which is similar to the ones available in the literature for covering set, equitable dominating set and so on. Using the idea of signless Laplacian, one can add this diagonal matrix to the usual adjacency matrix of G to obtain new matrix called independent transversal dominating adjacency matrix denoted by Ait(G). The sum of all the absolute values of eigenvalues of Ait(G) is called independent transversal dominating energy denoted by Eit(G). In the present paper, some spectral properties of Ait(G) are obtained. Some upper and lower bounds for the largest eigenvalue of Ait(G) and Eit(G) are derived. Eit(G) is obtained for some standard graphs.
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