Sharp bounds on the generalized distance spectral radius and generalized distance energy of strongly connected digraphs

College of Science, Northwest A&F University, Yangling, Shaanxi 712100, P.R. China

^* Corresponding author: xiyanxwg@163.com

Received: 1 March 2024
Accepted: 11 September 2024

Abstract

Let D(G) be the distance matrix of a strongly connected digraph G, Tr(G) be the diagonal matrix with vertex transmissions of G as diagonal entries. The generalized distance matrix D_β(G) of the strongly connected digraph G is defined as D_β(G) = βTr(G)+(1−β)D(G), for any real 0 ≤ β ≤ 1. The generalized distance spectral radius of G is the spectral radius of D_β(G). Let μ1β(G),μ2β(G),...,μnβ(G) be the eigenvalues of D_β(G), the generalized distance energy of the digraph G is EDβ(G)=Σi=1n|μiβ(G)−βW(G)n|, where W (G) is the sum of distances between all ordered pairs of vertices of G. In this paper, we obtain some sharp upper and lower bounds for the generalized distance spectral radius of G and characterize the extremal digraphs. Moreover, we also give some lower bounds on the generalized distance energy of digraphs.