Bounding the sum of powers of the Laplacian eigenvalues of graphs

• Published in: Applied Mathematics-A Journal of Chinese Universities Vol. 26; no. 2; pp. 142 - 150
• Main Authors: Chen, Xiao-dan, Qian, Jian-guo
• Format: Journal Article
• Language: English
• Published: Heidelberg SP Editorial Committee of Applied Mathematics - A Journal of Chinese Universities 01.06.2011
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; chromatic number; Kirchhoff index; connectivity; 05C50; the first Zagreb index; Laplacian eigenvalues
• ISSN: 1005-1031; 1993-0445
• DOI: 10.1007/s11766-011-2732-4

For a non-zero real number α , let s α ( G ) denote the sum of the α th power of the non-zero Laplacian eigenvalues of a graph G . In this paper, we establish a connection between s α ( G ) and the first Zagreb index in which the Hölder’s inequality plays a key role. By using this result, we present a lot of bounds of s α ( G ) for a connected (molecular) graph G in terms of its number of vertices (atoms) and edges (bonds). We also present other two bounds for s α ( G ) in terms of connectivity and chromatic number respectively, which generalize those results of Zhou and Trinajstić for the Kirchhoff index [B Zhou, N Trinajstić. A note on Kirchhoff index , Chem. Phys. Lett., 2008, 455: 120–123].