arXiv subjects
Publications and source records attributed to Germain Pastén.
Let $G$ be a graph with adjacency matrix $A(G)$ and Laplacian matrix $L(G)$. In 2024, Samanta \textit{et} \textit{al.} defined the convex linear combination of $A(G)$ and $L(G)$ as $B_α(G) = αA(G) + (1-α)L(G)$, for $α\in [0,1]$. This paper presents some results on the eigenvalues of $B_α(G)$ and their multiplicity when some sets of vertices satisfy certain conditions. Moreover, the positive semidefiniteness problem of $B_α(G)$ is studied.
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right],$ define the matrix $A_α\left(G\right) $ as \[ A_α\left(G\right) =αD\left(G\right) +(1-α)A\left(G\right) \] where $0\leqα\leq1$. This paper gives several results about the $A_α$-matrices of trees. In particular, it is shown that if $T_Δ$ is a tree of maximal degree $Δ,$ then the spectral radius of $A_α(T_Δ)$ satisfies the tight inequality \[ ρ(A_α(T_Δ))<αΔ+2(1-α)\sqrt{Δ-1}. \] This bound extends previous bounds of Godsil, Lovász, and Stevanović. The proof is based on some new results about the $A_α$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_α$ of general graphs are proved, implying tight bounds for paths and Bethe trees.