Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $

Tian, Gui-Xian ; Huang, Ting-Zhu

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 567-580

Résumé

Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,\ldots ,d_n)$. Denote $(^\alpha t)_i = \sum \nolimits _{j\colon i \sim j} {d_j^\alpha }$, $(^\alpha m)_i = {(^\alpha t)_i }/{d_i^\alpha }$ and $(^\alpha N)_i = \sum \nolimits _{j\colon i \sim j} {(^\alpha t)_j }$, where $\alpha $ is a real number. Denote by $\lambda _1(G)$ and $\mu _1(G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of $\lambda _1(G)$ and $\mu _1(G)$ in terms of $(^\alpha t)_i $, $(^\alpha m)_i $ and $(^\alpha N)_i $. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

Détail

DOI : 10.1007/s10587-012-0030-9

Classification : 05C50, 15A18
Keywords: graph; adjacency matrix; Laplacian matrix; spectral radius; bound