Sharp lower bounds for the Laplacian Estrada index of graphs

Let G be a simple graph on n vertices, and let lambda(1), lambda(2), & mldr;, lambda n be the Laplacian eigenvalues of G. The Laplacian Estrada index of G is defined as LEE(G) = & sum;(n )(i=1)e(lambda i). Consider a graph G with n >= 3 vertices, m edges, c connected components, and the largest Laplacian eigenvalue lambda(n). Let K-n, S-n, and K-p,K- q (p + q = n) denote the complete graph, the star graph, and the complete bipartite graph on n vertices, respectively. In this paper, we establish that LEE(G) >= ne(2m/n )+ c + e(lambda n )- (c+1)e(lambda n/c+1). Furthermore, we show that the equality holds if and only if G congruent to (K) over bar (n) (the complement of K-n), G congruent to boolean OR K-c-1 (i=1)1 boolean OR S(c+1 )if n = 2c, or G congruent to K(n/2, n/2 )if G is a connected graph on an even number of vertices. As a consequence of this lower bound, we derive sharp lower bounds for the Laplacian Estrada index of a graph, considering its well-known graph parameters. This leads to improvements to some previously known lower bounds for the Laplacian Estrada index of a graph. Notably, we establish a sharp lower bound for the Laplacian Estrada index of a graph in terms of its maximum vertex degree. As an application, we demonstrate that the lower bound for the Laplacian Estrada index presented by Khosravanirad in [A Lower Bound for Laplacian Estrada Index of a Graph, MATCH Commun Math Comput Chem. 2013;70:175-180.] is not complete. Consequently, we provide a complete version of this lower bound.