Some results on extremal spectral radius of hypergraph

For a $hypergraph$ $\mathcal{G}=(V, E)$ with a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as $(\mathcal {A}_{\mathcal{G}})_{ij}=\sum_{e\in E_{ij}}\frac{1}{|e| - 1}$, where $E_{ij} = \{e\in E\, |\, i, j \in e\}$. The $spectral$ $radius$ of a hypergraph $\mathcal{G}$, denoted by $\rho(\mathcal {G})$, is the maximum modulus among all eigenvalues of $\mathcal {A}_{\mathcal{G}}$. In this paper, we get a formula about the spectral radius which link the ordinary graph and the hypergraph, and represent some results on the spectral radius changing under some graphic structural perturbations. Among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices, the hypergraphs with the minimum, the second the minimum spectral radius are completely determined, respectively; among all $k$-uniform ($k\geq 3$) unicyclic hypergraphs with fixed number of vertices and fixed girth, the hypergraphs with the maximum spectral radius are completely determined; among all $k$-uniform ($k\geq 3$) $octopuslike$ hypergraphs with fixed number of vertices, the hypergraphs with the minimum spectral radius are completely determined. As well, for $k$-uniform ($k\geq 3$) $lollipop$ hypergraphs, we get that the spectral radius decreases with the girth increasing.Comment: arXiv admin note: substantial text overlap with arXiv:2306.10184, arXiv:2306.1602

The domination number and the least QQ-eigenvalue

A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the least eigenvalue of its signless Laplacian matrix. In this paper, for a nonbipartite graph with both order $n$ and domination number $\gamma$, we show that $n\geq 3\gamma-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $\gamma$. By investigating the relation between the domination number and the least $Q$-eigenvalue of a graph, we minimize the least $Q$-eigenvalue among all the nonbipartite graphs with given domination number.Comment: 13 pages, 3 figure