Eigenvalue bounds for the signless \(p\)-Laplacian (Q1753095)

Summary: We consider the signless \(p\)-Laplacian \(Q_p\) of a graph, a generalisation of the quadratic form of the signless Laplacian matrix (the case \(p=2\)). In analogy to Rayleigh's principle the minimum and maximum of \(Q_p\) on the \(p\)-norm unit sphere are called its smallest and largest eigenvalues, respectively. We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a graph parameter related to the bipartiteness. The latter result generalises bounds by \textit{M. Desai} and \textit{V. Rao} [J. Graph Theory 18, No. 2, 181--194 (1994; Zbl 0792.05096)] and, interestingly, at \(p=1\) upper and lower bounds coincide.