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 $(d_1,d_2,\dots ,d_n)$. Denote ${(}^{\alpha }t{)}_i={\sum }_{j:i\sim j}{d}_j^{\alpha }$, ${(}^{\alpha }m{)}_i={(}^{\alpha }t{)}_i/d_i^{\alpha }$ and ${(}^{\alpha }N{)}_i={\sum }_{j:i\sim j}{(}^{\alpha }t{)}_j$, 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
Mots-clés : graph; adjacency matrix; Laplacian matrix; spectral radius; bound