The Lollipop Graph is Determined by its Spectrum

Abstract

An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. A private communication of Behruz Tayfeh-Rezaie pointed out that an even lollipop with a cycle of length at least $6$ is determined by its spectrum but the result for lollipops with a cycle of length $4$ is still unknown. We give an unified proof for lollipops with a cycle of length not equal to $4$, generalize it for lollipops with a cycle of length $4$ and therefore answer the question. Our proof is essentially based on a method of counting closed walks.

Laplacian Spectral Characterization of Some Unicyclic Graphs

“…Here, we describe a classic method to count the number of closed walks of a given length in a graph (see [2,13,14]). For a graph , ( ) stands for the number of closed walks of length in and ( ) stands for the number of subgraphs of which are isomorphic to graph .…”

“…Haemers et al [1] proved that ( ; , 1 ) is determined by its A-spectrum when is odd, and all ( ; , 1 ) are determined by their L-spectra. It is also known that ( ; , 1 ) is determined by its Aspectrum when is even [2]. Liu et al [3] proved that ( ; , 1 , 2 ) is determined by its L-spectrum.…”

On the spectral determinations of the connected multicone graphs

“…So far numerous examples of cospectral but non-isomorphic graphs have been constructed by interesting techniques such as Seidel switching, Godsil-McKay switching, Sunada or Schwenk method. For more information, one may see [10,28,29] have been reported to be determined by their spectra (DS, for short) (see [11,13,15,18,22,27,32,31] and the references cited in them). Recently Wei Wang and Cheng-Xian Xu have developed a new method in [32] to show that many graphs are determined by their spectrum and the spectrum of their complement.…”

On the spectral characterization of kite graphs

“…But it is difficult to show that a given graph is DS. Up to now, only few graphs are proved to be DS [2][3][4][5][6][7][9][10][11][12][13]15]. Recently, some papers have been appeared that focus on some special graphs (oftenly under some conditions) and prove that these special graphs are DS or non-DS [2-4, 7, 9-13, 15].…”