Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector

Abstract

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

New Bounds for the Generalized Distance Spectral Radius/Energy of Graphs

“…We know that Tr i � k and T i � k 2 . According to Lemma 1, k is the largest eigenvalue of D α (G), and ρ(D α (G)) � k, so the equality holds in (15). On the contrary, if inequality ( 15) is equal, then x 1 � x 2 � .…”

“…In particular, they showed that this upper bound is better than the upper bound obtained by Cui et al In [14], the authors established the relations between the smallest eigenvalues of D α (G), D(G), and D Q (G) and obtained sharp bounds for the smallest eigenvalue λ n of D α (G) in terms of various graph parameters. In [12,[15][16][17], Alhevaz et al established some new sharp bounds for the generalized distance spectral radius of G by using diferent graph parameters and characterized the extremal graphs. Ten, they obtained new bounds for the k-th generalized distance eigenvalue.…”