Some Graphs Determined By Their Laplacian Spectra

In this paper, we mainly study a hot topic in the Spectral Graph Theory:the spectral characterization problem. In [1,2], Zhang and Liu have proved that the wheel graph Cn1▽K1 when n1≠6 and the multi-fan graph (Pn1∪Pn2∪…∪Pns)▽K1 are determined by their Laplacian spectra. On the basis of the above two theorems, we generalize the results from K1 to Kκ. whereκis a general integer. Furthermore, we have discussed the Laplcian spectral characterization of the generalized wheel graph and the generalized multi-fan graph. We also give a group of Laplacian cospectral graphs. Our main result is:1. First, we have proved that the CniVKk is determined by its Laplacian spectrum except n1=6. In our proof, we discuss our problem on two cases that n1 is odd or even. We also point that when n1= 6, Cn1▽Kκhas Laplacian cospectral graph.2. We have also proved that (Pn1∪Pn2∪…∪Pns)▽Kκis determined by its Laplacian spectrum.3. Then we give a cospectral graph of wheel graph and provide a conjecture that all odd wheel graphs are determined by their adjacency spectra.4. At last, we provide some graphs determined by their signless Laplacian spectra.