Title

Graph functions maximized on a path

Abstract

Given a connected graph G of order n and a nonnegative symmetric matrix A=[ai,j] of order n, define the function FA(G) as FA(G)=Σ1≤iG(i,j)ai,j, where dG(i,j) denotes the distance between the vertices i and j in G. In this note it is shown that FA(G)≤FA(P) for some path of order n. Moreover, if each row of A has at most one zero off-diagonal entry, then FA(G)A(P) for some path of order n, unless G itself is a path. In particular, this result implies two conjectures of Aouchiche and Hansen: the spectral radius of the distance Laplacian of a connected graph G of order n is maximal if and only if G is a path;the spectral radius of the distance signless Laplacian of a connected graph G of order n is maximal if and only if G is a path.

Publication Title

Linear Algebra and Its Applications

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