Abstract
A positive eigenvector corresponding to the largest eigenvalue of a symmetric, non-negative and irreducible matrix is known as principal eigenvector of the matrix. For a real number p, 1 ≤ p< ∞, a principal eigenvector (y1,y2,…,yn)T is said to be p-norm normalized if (∑i=1nyip)1p=1. The distance matrix of a simple connected graph G is D(G) = (dij) , where dij is the distance between ith and jth vertices of G. The distance signless Laplacian matrix of the graph G is DQ(G) = D(G) + Tr(G) , where Tr(G) is a diagonal matrix whose ith diagonal entry is the transmission of the vertex i in G. In this paper we find some upper and lower bounds of the maximal and minimal entry of the p-normalized principal eigenvector for the distance matrix and distance signless Laplacian matrix of a graph and show that transmission regular graphs are extremal for all these bounds. We also compare these bounds with the bounds in the literature.