Abstract
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order $n$; (2) the maximum independence number of an almost self-centered graph of order $n$ and radius $r$; (3) the minimum order of a $k$-regular almost self-centered graph; (4) the maximum size of an almost peripheral graph of order $n$; (5) possible maximum degrees of an almost peripheral graph of order $n$ and (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order $n$ with maximum degree $n-4$ which is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.
Funding Statement
This research was supported by the NSFC grant 11671148 and Science and Technology Commission of Shanghai Municipality (STCSM) grant 18dz2271000.
Citation
Yanan Hu. Xingzhi Zhan. "On Almost Self-centered Graphs and Almost Peripheral Graphs." Taiwanese J. Math. 26 (5) 887 - 901, October, 2022. https://doi.org/10.11650/tjm/220401
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