The Laplacian energy of random graphs

Details The Laplacian energy of random graphs The Laplacian energy of random graphs

2009-06-25

arxiv.org/abs/0906.4636v4

Mathematics

Combinatorics

14 pages

Scientific paper

Gutman {\it et al.} introduced the concepts of energy $\En(G)$ and Laplacian energy $\EnL(G)$ for a simple graph $G$, and furthermore, they proposed a conjecture that for every graph $G$, $\En(G)$ is not more than $\EnL(G)$. Unfortunately, the conjecture turns out to be incorrect since Liu {\it et al.} and Stevanovi\'c {\it et al.} constructed counterexamples. However, So {\it et al.} verified the conjecture for bipartite graphs. In the present paper, we obtain, for a random graph, the lower and upper bounds of the Laplacian energy, and show that the conjecture is true for almost all graphs.

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