On Laplacian–Energy–Like Invariant and Kirchhoff Index S. Pirzada 1 , Hilal A. Ganie 1 , Ivan Gutman 2,3 1 Department of Mathematics, University of Kashmir, Srinagar, India pirzadasd@kashmiruniversity.ac.in , hilahmad1119kt@gmail.com 2 Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia gutman@kg.ac.rs 3 State University of Novi Pazar, Novi Pazar, Serbia (Received October 13, 2014) Abstract For a simple connected graph G of order n, the Laplacian–energy–like invariant and the Kirchhoff index are calculated by LEL(G)= n−1 ∑ i=1 √ μ i and Kf (G)= n n−1 ∑ i=1 1/μ i , respectively, where μ 1 ,μ 2 ,...,μ n−1 ,μ n = 0 are the Laplacian eigenvalues of G. We obtain a sharp upper bound for Kf and a sharp lower bound for LEL. Further, we obtain upper and lower bounds for LEL and Kf for graphs C(G) (the clique–inserted graph or para-line graph), R(G) (obtained by changing each edge of G into a triangle), and H(G) (obtained by inserting a new vertex on each edge of G and by joining two new vertices if they lie on adjacent edges of G), as well as for the line graph of a semiregular graph. 1 Introduction Let G be a finite, undirected, simple graph with n vertices and m edges, having vertex set V (G)= {v 1 ,v 2 ,...,v n }. The adjacency matrix A(G)=(a ij ) of G is the (0, 1)-square matrix of order n whose (i,j )-entry is equal to one if v i is adjacent to v j and equal to zero otherwise. The eigenvalues of A(G) will be labeled as λ n ≤···≤ λ 2 ≤ λ 1 . MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 73 (2015) 41-59 ISSN 0340 - 6253