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Keywords:
Laplacian matrix; Kirchhoff index; hypertriangle; uniform Fano plane; first Zagreb index
Laplacian matrix; Kirchhoff index; hypertriangle; uniform Fano plane; first Zagreb index
Summary:
Let $\mathcal {H}$ be a connected $k$-uniform hypergraph on $n$ vertices and $m$ hyperedges. K. Feng, W. Li (1996) introduced an adjacency matrix $\mathcal {A}(\mathcal {H})$ for hypergraphs. We consider the corresponding Laplacian matrix. We extend the concept of the Kirchhoff index to connected hypergraphs. We compute the Kirchhoff index for uniform complete, uniform complete bipartite, hypertriangle, and uniform Fano plane. A hypergraph is the Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. In the process, three different classes of hypergraphs with Laplacian integral are provided. We show that the Kirchhoff index of any connected $k$-uniform hypergraph $\mathcal {H}$ is at least $(n-1)/\binom {n-2}{k-2}$ and the equality holds if and only if $\mathcal {H}$ is a $k$-uniform complete hypergraph. We also obtain some bounds for the Kirchhoff index in terms of hypergraph invariants such as the number of vertices, number of hyperedges, and first Zagreb index.
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