A peacock graph \(PG=(i, j, k\,; b_1, b_2,\ldots ,b_s)\), where \(i,j,k\ge 3\) is a graph consisting of three cycles \(C_i\), \(C_j\), \(C_k\) and \(s\,(\ge 1)\) 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. 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. If \(s=1\), then the peacock graph is called clover graph. In Wang and Wang (Linear Multilinear Algebra 63(12):2396–2405, 2015), it was proved that all clover graphs are DLS. In this paper, we generalize this result and show that all peacock graphs are DLS.

中文翻译：

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孔雀图由其拉普拉斯谱确定

孔雀图\（PG =（i，j，k \ ,; b_1，b_2，\ ldots，b_s）\），其中\（i，j，k \ ge 3 \）是由三个周期\（ C_i \），\（C_j \），\（C_k \）和\（s \，（\ ge 1）\）路径\（P_ {b_1 + 1}，P_ {b_2 + 1}，\ ldots，P_ { b_s + 1} \）相交在一个顶点上的单个顶点相交。如果每个具有相同拉普拉斯光谱的图都与G同构，则称图G是由其拉普拉斯矩阵（简称LPS）的光谱确定的。如果\（s = 1 \），则孔雀图称为三叶草图。在Wang和Wang中（线性多线性代数63（12）：2396-2405，2015），证明所有三叶草图都是DLS。在本文中，我们对该结果进行了概括，并表明所有孔雀图都是DLS。