GRAPHS WHICH ARE DETERMINED BY THEIR SPECTRUM

Abstract

It is well-known that the problem of spectral characterization is related to the H\"uckel theory from Chemistry. E. R. van Dam and W. H. Haemers $ [11] $ conjectured almost all graphs are determined by their spectra. Nevertheless, the set of graphs which are known to be determined by their spectra is small. Hence discovering infinite classes of graphs that are determined by their spectra can be an interesting problem and helps reinforce this conjecture. The main aim of this work is to characterize new classes of graphs that are known as multicone graphs. In this work, it is shown that any graph cospectral with multicone graphs $ K_w\bigtriangledown GQ(2,1) $ or $ K_w\bigtriangledown GQ(2,2) $ is determined by its adjacency spectra, where $ GQ(2,1) $ and $ GQ(2,2) $ denote the strongly regular graphs that are known as the generalized quadrangle graphs. Also, we prove that these graphs are determined by their Laplacian spectrum. Moreover, we propose four conjectures for further reseache in this topic.

Keywords

• Adjacency spectrum,Laplacian spectrum,Determined by their spectra,generalized quadrangle

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