# Graphs with small diameter determined by their *D*-spectra

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## Abstract

Let *G* be a connected graph with vertex set *V(G)* = {*v*_{1}, *v*_{2},..., *v*_{ n }}. The distance matrix *D*(*G*) = (*d*_{ ij })_{n×n} is the matrix indexed by the vertices of *G*, where *d*_{ ij } denotes the distance between the vertices *v*_{ i } and *v*_{ j }. Suppose that λ_{1}(*D*) ≥ λ_{2}(*D*) ≥... ≥ λ_{ n }(*D*) are the distance spectrum of *G*. The graph *G* is said to be determined by its *D*-spectrum if with respect to the distance matrix *D*(*G*), any graph having the same spectrum as *G* is isomorphic to *G*. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their *D*-spectra.

## Keywords

distance spectrum distance characteristic polynomial distance characteristic polynomial*D*-spectrum deter- mined by its

*D*-spectrum

## MSC 2010

05C50## Preview

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