Document Type : Research Paper

**Author**

Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71457-44776, Iran.

**Abstract**

A setosa graph $SG(e,f,g,h,d;b_1,b_2,\ldots,b_s)$ is a graph consisting of five cycles and $s(\geq 1)$ paths $P_{b_1+1}, P_{b_2+1},\ldots,P_{b_s+1}$ intersecting in a single vertex that all meet in one vertex, where $b_i\geq1$ (for $i=1,\ldots,s$) and $e,f,g,h,d\geq 3$ denote the length of the cycles $C_e$, $C_f$, $C_g$, $C_h$ and $C_d$, respectively. Two graphs $G$ and $H$ are $L$-*cospectral* if they have the same Laplacian spectrum. A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS, for short) if every graph with the same Laplacian spectrum is isomorphic to $G$. In this paper we prove that if $H$ is a $L$-cospectral graph with a setosa graph $G$, then $H$ is also a setosa graph and the degree sequence of $G$ and $H$ are the same. We conjecture that all setosa graphs are DLS.

**Keywords**

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February 2023

Pages 39-46