Abstract:

The spectral extremal problem and graph are hot issues in the study of graph theory nowadays. Scholars are keen to study the extremal graphs attaining the maximum or minimum spectral radius of graph classes. In this paper, the extremal graph of the second largest unsigned Laplacian spectral radius of a supertree with diameter of $4$ is characterized. Let $\mathbb{S}(m, 4, k)$ be the set of $k$-uniform supertree with $m$ edges and diameter $4$, and $S_3(m, 4, k)$ be the $k$-uniform supertree obtained from a loose path $v_1e_1v_2e_2v_3e_3v_4e_4v_5$ with length $4$ by attaching $m-4$ edges at vertex $v_4$. In this paper, firstly, introducing the definition of edge-shifting operation and related theorems. Then, according to edge-shifting operation, we find $S_3(m, 4, k)$ is the graph with the second largest signless Laplacian spectral radius in $\mathbb{S}(m, 4, k)$.

Key words: uniform hypergraph, supertree, signless Laplacian spectral radius, pendent edge