Some Bounds for the Incidence Q-Spectral Radius of Uniform Hypergraphs

Abstract

For a hypergraph H, let B(H) be its incidence matrix. The signless Laplacian matrix $Q(H)=B(H)B(H)^T$, whose (i, j)-element is exactly the number of edges containing vertices $v_i$ and $v_j$ for $i\ne j$ and (i, i)-element is exactly the degree of vertex $v_i$. The incidence Q-tensor ${\mathcal {Q}}^*=B(H){\mathcal {I}}B(H)^T$, whose $(i_1,i_2,\ldots ,i_k)$-entry of ${\mathcal {Q}}^*$ is exactly equal to the number of edges e of H satisfying $i_t\in e$ for all $t\in [k]$. Obviously, we can see more complete structural properties of H from ${\mathcal {Q}}^*$ than Q(H). Up to now, few tensors whose entries are directly related to structural properties of the corresponding hypergraphs. In this regard, we believe that it deserve to study the structural properties of hypergraphs though this tensor. In this paper, we will study some upper bounds on the spectral radius of ${\mathcal {Q}}^*$ by some parameters on hypergraphs.