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Abstract
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A graph $G$ is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum as $G$. Let $k$, $t_i$ ($1\leq i\leq k$), and $s$ be natural numbers. A path-friendship graph, $G_{s, t_1, \dots, t_k}$, is a graph of order $n=2s+t_1+\cdots+t_k+1$ which consists of $s$ triangles and $k$ paths of lengths $t_1, t_2,\ldots, t_k $ sharing a common vertex. In this paper, we show that these graphs are DQS and using this result, we respond to a conjecture in [F. Wen, Q. Huang, X. Huang and F. Liu, The spectral characterization of wind-wheel graphs, Indian J. Pure Appl. Math. 46 (2015), 613--631].
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