Regular factors of regular graphs from eigenvalues.

Lu, Hongliang

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A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.

Remarks on spectral radius and Laplacian eigenvalues of a graph

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Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_{1}, d_{2}, \dots, d_{n})$ , where $d_{1} \geq d_{2} \geq \dots \geq d_{n}$ . The spectral radius and the largest Laplacian eigenvalue are denoted by $ρ (G)$ and $μ (G)$ , respectively. We determine the graphs with $ρ (G) = \frac{d_{n} - 1}{2} + \sqrt{2 m - n d_{n} + \frac{{(d_{n} + 1)}^{2}}{4}}$ and the graphs with $d_{n} \geq 1$ and $μ (G) = d_{n} + \frac{1}{2} + \sqrt{\sum_{i = 1}^{n} d_{i} (d_{i} - d_{n}) + {(d_{n} - \frac{1}{2})}^{2}} .$ We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.