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arXiv · 1809.01901

Extremal graphs for vertex-degree-based invariants with given degree sequences

Muhuo Liu·Kexiang Xu·Xiao-Dong Zhang
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Abstract

For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $\pi=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invariants based on degree are included.

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Muhuo Liu, Kexiang Xu, Xiao-Dong Zhang. 2018-09-06. Extremal graphs for vertex-degree-based invariants with given degree sequences. https://arxiv.org/abs/1809.01901

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