Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph , a vertex subset is called a maximum generalized 4-independent set of if the induced subgraph dose not contain a 4-tree as its subgraph, and the subset has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of . In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the -vertex graphs having the minimum spectral radius with generalized 4-independence number , where . Finally, we identify all the connected -vertex graphs with generalized 4-independence number having the minimum spectral radius.

中文翻译：

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给定阶数和广义4-独立数图的谱极值


表征给定类图中具有最大或最小谱半径的图是谱极值图论中的一个经典问题，最初由 Brualdi 和 Solheid 提出。给定一个图 ，如果导出子图不包含 4 树作为其子图，并且该子集具有最大基数，则顶点子集称为最大广义 4 独立集。最大广义四独立集的基数称为 的广义四独立数。在本文中，我们首先确定在所有具有固定阶和广义 4-独立数的连通图（分别为二分图、树）中具有最大谱半径的连通图（分别为二分图、树），此外，我们建立具有固定顺序的树的广义 4 独立数的下界。其次，我们用广义 4-独立数 来描述具有最小谱半径的所有 - 顶点图的结构，其中 。最后，我们识别所有具有最小谱半径的广义 4-独立数的连通顶点图。