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Abstract: Let G be a connected graph with vertex set V(G)=\{v_1,v_2,\ldots,v_n\}. The distance matrix D(G)=(d_{ij})_{n\times 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 \lambda_1(D)\geq\lambda_2(D)\geq\cdots\geq\lambda_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. |
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