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Remarks on the bounds of graph energy in terms of vertex cover number or matching number

Xiaodan Chen; Xiaoqian Liu — 2021

Czechoslovak Mathematical Journal

We give a novel upper bound on graph energy in terms of the vertex cover number, and present a complete characterization of the graphs whose energy equals twice their matching number.

The extremal irregularity of connected graphs with given number of pendant vertices

Xiaoqian Liu; Xiaodan Chen; Junli Hu; Qiuyun Zhu — 2022

Czechoslovak Mathematical Journal

The irregularity of a graph $G = (V, E)$ is defined as the sum of imbalances $| d_{u} - d_{v} |$ over all edges $u v \in E$ , where $d_{u}$ denotes the degree of the vertex $u$ in $G$ . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with $n$ vertices and $p$ pendant vertices ( $1 \leq p \leq n - 1$ ), and characterize the corresponding extremal graphs.

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