Characterizing properties of permanental polynomials of lollipop graphs

A lollipop graph, denoted by, is a graph obtained by appending a cycle to a pendant vertex of a path. It is known that the lollipop graphs are determined by their spectra [Haemers WH, Liu X, Zhang Y. Spectral characterizations of lollipop graphs. Linear Algebra Appl. 2008;428:2415–2423; Boulet R, Jouve B. The lollipop graph is determined by its spectrum. Electron. J. Combin. 2008;15:#R74]. In this paper, we consider whether the lollipop graphs can be determined by the permanental polynomial. We show that () and are determined by their permanental polynomials when restricted to connected graphs, and in general () cannot be determined by their permanental polynomials even when restricted to connected graphs. In particular, we find disconnected copermanental mates of for and connected copermanental mates of,,, and for, respectively.

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