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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2411.09963 |
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| _version_ | 1866909391129673728 |
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| author | Ye, Jiachang |
| author_facet | Ye, Jiachang |
| contents | Let $λ_{i}(G)$ be the $i$-th largest Laplacian eigenvalues of graph $G$, where $1\le i\le |V(G)|$. Liu, Yuan, You and Chen [Discrete Math., 341 (2018) 2969--2976] raised the problem for ``Which cospectral graphs have same degree sequences". In this paper, let $W_3$ and $W_5$ be the two graphs as shown in Fig. 2 and let $G$ be a connected graph with $n\ge 18$ vertices. We shall show that:
$(1)$ If $λ_{2}(G)<5<n-1<λ_{1}(G)$, $λ_{1}(G) \notin \{λ_{1}(W_3),λ_{1}(W_5)\}$ and $H$ is Laplacian cospectral with $G$, then $H$ must have the same degree sequence with $G$;
$(2)$ If $λ_2(G)\le 4.7<n-2< λ_1(G)$, and $H$ is Laplacian cospectral with $G$, then $H$ must have the same degree sequence with $G$.
The former result easily leads to the unique theorem result of [Discrete Math., 308 (2008) 4267--4271], that is: Every multi-fan graph $K_1\vee (P_{l_1}\cup P_{l_1}\cup\cdots \cup P_{l_t})$ is determined by the Laplacian spectrum. Moreover, it can also deduce a new conclusion: $K_1\vee (P_{l_1}\cup P_{l_1}\cup\cdots \cup P_{l_t}\cup C_{s_1}\cup C_{s_2}\cup\cdots \cup C_{s_k})$ $(t\ge 1, k\ge 1)$ is determined by the Laplacian spectrum if the graph order $n\ge 18$ and each $s_i$ $(i=1,2,\ldots, k)$ is odd. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09963 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Which $L$-cospectral graphs have same degree sequences Ye, Jiachang Combinatorics Let $λ_{i}(G)$ be the $i$-th largest Laplacian eigenvalues of graph $G$, where $1\le i\le |V(G)|$. Liu, Yuan, You and Chen [Discrete Math., 341 (2018) 2969--2976] raised the problem for ``Which cospectral graphs have same degree sequences". In this paper, let $W_3$ and $W_5$ be the two graphs as shown in Fig. 2 and let $G$ be a connected graph with $n\ge 18$ vertices. We shall show that: $(1)$ If $λ_{2}(G)<5<n-1<λ_{1}(G)$, $λ_{1}(G) \notin \{λ_{1}(W_3),λ_{1}(W_5)\}$ and $H$ is Laplacian cospectral with $G$, then $H$ must have the same degree sequence with $G$; $(2)$ If $λ_2(G)\le 4.7<n-2< λ_1(G)$, and $H$ is Laplacian cospectral with $G$, then $H$ must have the same degree sequence with $G$. The former result easily leads to the unique theorem result of [Discrete Math., 308 (2008) 4267--4271], that is: Every multi-fan graph $K_1\vee (P_{l_1}\cup P_{l_1}\cup\cdots \cup P_{l_t})$ is determined by the Laplacian spectrum. Moreover, it can also deduce a new conclusion: $K_1\vee (P_{l_1}\cup P_{l_1}\cup\cdots \cup P_{l_t}\cup C_{s_1}\cup C_{s_2}\cup\cdots \cup C_{s_k})$ $(t\ge 1, k\ge 1)$ is determined by the Laplacian spectrum if the graph order $n\ge 18$ and each $s_i$ $(i=1,2,\ldots, k)$ is odd. |
| title | Which $L$-cospectral graphs have same degree sequences |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.09963 |