Merging the A- and Q-spectral theories

Let $G$ be a graph with adjacency matrix $A\left( G\right)$, and let $D\left( G\right)$ be the diagonal matrix of the degrees of $G.$ The signless Laplacian $Q\left( G\right)$ of $G$ is defined as $Q\left( G\right) :=A\left( G\right) +D\left( G\right)$. Cvetkovi\'{c} called the study of the adjacency matrix the $A$% \textit{-spectral theory}, and the study of the signless Laplacian--the $Q$\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left( G\right)$ into $Q\left( G\right)$ in this paper it is suggested to study the convex linear combinations $A_{\alpha }\left( G\right)$ of $A\left( G\right)$ and $D\left( G\right)$ defined by $$A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1.$$ This study sheds new light on $A\left( G\right)$ and $Q\left( G\right)$, and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.Comment: 26 page

A Sharp upper bound for the spectral radius of a nonnegative matrix and applications

In this paper, we obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. These results are new or generalize some known results.Comment: 16 pages in Czechoslovak Math. J., 2016. arXiv admin note: text overlap with arXiv:1507.0705

On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor signed graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles. We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page