Seyed Ahmad Mojallal

On Laplacian energy of graphs

Abstract Let G be a graph with n vertices and m edges. Also let μ 1 , μ 2 , … , μ n − 1 , μ n = 0 be the eigenvalues of the Laplacian matrix of graph G . The Laplacian energy of the graph G is defined as L E = L E ( G ) = ∑ i = 1 n | μ i − 2 m n | . In this paper, we present some lower and upper bounds for L E of graph G in terms of n , the number of edges m and the maximum degree Δ . Also we give a Nordhaus–Gaddum-type result for Laplacian energy of graphs. Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs.

On energy and Laplacian energy of bipartite graphs

Let G be a bipartite graph of order n with m edges. The energy E ( G ) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors established lower and upper bounds for E ( G ) in terms of n, m, and det A . Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized.

On Laplacian energy in terms of graph invariants

For G being a graph with n vertices and m edges, and with Laplacian eigenvalues µ 1 ? µ 2 ? ? ? µ n - 1 ? µ n = 0 , the Laplacian energy is defined as L E = ? i = 1 n | µ i - 2 m / n | . Let ? be the largest positive integer such that µ? ? 2m/n. We characterize the graphs satisfying ? = n - 1 . Using this, we obtain lower bounds for LE in terms of n, m, and the first Zagreb index. In addition, we present some upper bounds for LE in terms of graph invariants such as n, m, maximum degree, vertex cover number, and spanning tree packing number.

Spectral properties of Pascal graphs

Abstract We obtain spectral properties of the Pascal graphs by exploring its spectral graph invariants such as the algebraic connectivity, the first three largest Laplacian eigenvalues and the nullity. Some open problems pertaining to the Pascal graphs are given.

Extremal Laplacian energy of threshold graphs

Let G be a connected threshold graph of order n with m edges and trace T. In this paper we give a lower bound on Laplacian energy in terms of n, m and T of G. From this we determine the threshold graphs with the first four minimal Laplacian energies. Moreover, we obtain the threshold graphs with the largest and the second largest Laplacian energies.