Journal of Combinatorial Optimization ( IF 0.9 ) Pub Date : 2024-06-28 , DOI: 10.1007/s10878-024-01182-2 Hongying Lin , Bo Zhou
Let G be a k-uniform hypergraph with vertex set [n] and edge set E(G), where \(k\ge 2\). For \(i\in [n]\), \(d_i\) denotes the degree of vertex i in G. The ABC spectral radius of G is
$$\begin{aligned} \max \left\{ k\sum _{e\in E(G)}\root k \of {\dfrac{\sum _{i\in e}d_{i} -k}{\prod _{i\in e}d_{i}}}\prod _{i\in e}x_i: \textbf{x}\in {\mathbb {R}}_+^n, \sum _{i=1}^nx_i^k=1\right\} . \end{aligned}$$We give tight lower and upper bounds for the ABC spectral radius, and determine the maximum ABC spectral radii of uniform hypertrees, uniform non-hyperstar hypertrees and uniform non-power hypertrees of given size, as well as the maximum ABC spectral radii of uniform unicyclic hypergraphs and linear uniform unicyclic hypergraphs of given size, respectively. We also characterize those uniform hypergraphs for which the maxima for the ABC spectral radii are actually attained in all cases.
中文翻译：
关于均匀超图的ABC谱半径
设 G 是具有顶点集 [n] 和边集 E(G) 的 k 均匀超图，其中 \(k\ge 2\)。对于\(i\in [n]\)，\(d_i\)表示G中顶点i的度数。G的ABC谱半径为
$$\begin{对齐} \max \left\{ k\sum _{e\in E(G)}\root k \of {\dfrac{\sum _{i\in e}d_{i} -k }{\prod _{i\in e}d_{i}}}\prod _{i\in e}x_i: \textbf{x}\in {\mathbb {R}}_+^n, \sum _ {i=1}^nx_i^k=1\right\} 。 \end{对齐}$$
我们给出了 ABC 谱半径的严格下界和上限，并确定了给定大小的均匀超树、均匀非超星超树和均匀非幂超树的最大 ABC 谱半径，以及均匀单环的最大 ABC 谱半径分别是给定大小的超图和线性均匀单环超图。我们还描述了那些在所有情况下实际上都达到 ABC 谱半径最大值的均匀超图。
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