Largest adjacency, signless Laplacian, and Laplacian H-eigenvalues of loose paths

Abstract

We investigate k-uniform loose paths. We show that the largest Heigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l\ge \mathrm{3}, we show that the largest H-eigenvalue of its adjacency tensor is {((\mathrm{1}+\sqrt{\mathrm{5}})/\mathrm{2})}^{\mathrm{2}/k} when l=\mathrm{3} and \lambda (A)={\mathrm{3}}^{\mathrm{1}/k} when l=\mathrm{4}, respectively. For the case of l\ge \mathrm{5}, we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval (2, 3) when l\ge \mathrm{5}. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.

Key words： H-eigenvalue; hypergraph; adjacency tensor; signless Laplacian tensor; Laplacian tensor; loose path

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