Abstract
The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. In this paper, we give several less restricted graft transformations that decrease the distance spectral radius, and determine the unique graph with minimum distance spectral radius among home-omorphically irreducible unicylic graphs on n ≥ 6 vertices, and the unique tree with minimum distance spectral radius among trees on n vertices with given number of vertices of degree two, respectively.
Highlights
We consider simple, finite, undirected and connected graphs
A homeomorphically irreducible unicylic graph is a unicylic graph with no vertex of degree two
The distance spectrum of G is the spectrum of the distance matrix of G, defined as the n by n symmetric matrix D(G) = (dG(u, v))u,v∈V (G)
Summary
Finite, undirected and connected graphs. For u ∈ V (G), let NG(u) be the set of neighbors of u in G. Lin and Zhou [8] determined the trees with maximum distance spectral radius among trees on n vertices with given number of vertices of degree two. We propose some graft transformations with less restricted conditions that decrease the distance spectral radius, and as applications, we identify the unique graphs that minimize the distance spectral radius among homeomorphically irreducible unicylic graphs on n ≥ 6 vertices, and among trees on n vertices with given number of vertices of degree two, respectively
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