Abstract
In this paper we express the distance spectrum of graphs with small diameter in terms of the eigenvalues of their adjacency matrix. We also compute the distance energy of particular types of graph and determine a sequence of infinite families of distance equienergetic graphs.
Highlights
For a simple graph G with vertex set V (G) and order n = |V (G)|, the characteristic polynomial ΦG is defined as the characteristic polynomial of its adjacency matrix AG
We present two general results that make a connection between the polynomials ΨG and ΦG of a graph with diameter two and a bipartite semiregular graph with diameter three
We express the distance eigenvalues in terms of the eigenvalues of particular types of graph specified in the corresponding subsections
Summary
The eigenvalues of G, λ1(G) ≥ λ2(G) ≥ · · · ≥ λn(G), are just the roots of ΦG, and the spectrum of G, denoted by ΣG, is the multiset of its eigenvalues. The distance eigenvalues (for short D-eigenvalues) of G are the roots of ΨG They form the multiset called the D-spectrum of G. Distance energy DE(G)) is defined as the sum of the absolute values of the eigenvalues We compute the D-spectrum of particular graphs with diameter at most four For diameter two, these are graphs with exactly two main eigenvalues; for diameter three – bipartite semiregular graphs; for diameter four – bipartite regular incidence graphs of two-class symmetric partial incomplete block designs. The last is followed by a sequence of examples considering infinite families of distance equienergetic graphs.
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