On the inverse sum indeg energy of trees

Let G be a graph of order n and d(i) be the degree of the vertex v(i), for i = 1, 2, ..., n. The ISI matrix of G is the square matrix of order n whose (i, j)-entry is equal to d(i)d(j)/d(i)+d(j) if v(i) is adjacent to v(j), and zero otherwise. Let mu(1) > mu(2) > ... > mu(n), be the eigenvalues of ISI matrix. The ISI energy of a graph G, denoted by xi(ISI)(G), is defined as the sum of the absolute values of the eigenvalues of ISI matrix. In this paper, we prove that the star has the minimum ISI energy among trees.