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

Abstract

We investigate k-uniform loose paths. We show that the largest H-eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l ⩾ 3, we show that the largest H-eigenvalue of its adjacency tensor is \({\left( {{\raise0.7ex\hbox{${\left( {1 + \sqrt 5 } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {1 + \sqrt 5 } \right)} 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} \right)^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 k}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$k$}}}}\) when l = 3 and λ(A) = 31/k when l = 4, respectively. For the case of l ⩾ 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 ⩾ 5. Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4.