On graphs with three distinct Laplacian eigenvalues |
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Authors: | Wang Yi Fan Yizheng Tan Yingying |
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Affiliation: | (1) School of Math. and Comput. Sci., Anhui Univ., Hefei, 230039, China;(2) Dept. of Math. and Phys., Anhui Institute of Architecture and Industry, Hefei, 230022, China |
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Abstract: | In this paper, an equivalent condition of a graph G with t (2 ≤ t ≤ n) distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular (necessarily be strongly regular), an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral. |
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Keywords: | Laplacian matrix spectrum Laplacian integral strongly regular graph. |
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