On the Laplacian coefficients and Laplacian-like energy of bicyclic graphs

Abstract

Let be the characteristic polynomial of Laplacian matrix of an n-vertex graph G. We present three transforms on graphs that decrease all Laplacian coefficients c _k (G), then we characterize the graphs with the minimal Laplacian-like energy, which is the sum of square roots of all roots on φ(G, λ), in the set of all bicyclic graphs with fixed order and matching number. Furthermore, we determine the graphs with the smallest and the second smallest Laplacian-like energy among all n-vertex connected bicyclic graphs except B _n , where B _n is the graph obtained from a four-vertex cycle C ₄ by adding an edge joining two non-adjacent vertices and adding n − 4 pendant edges to a vertex of degree 3.

Keywords:

• Laplacian coefficient
• Laplacian-like energy
• spanning forest
• bicyclic graph
• matching

AMS Subject Classification:

• 05C50

Acknowledgements

The author expresses his sincere gratitude to the referees for a very careful reading of this article and for all their insightful comments and valuable suggestions, which led to improving this article. This research was supported by the National Natural Science Foundation of China (10871204) and the Fundamental Research Funds for the Central Universities (09CX04003A).