{"title":"有向图的拉普拉斯特征值的k个最大绝对值之和","authors":"Xiuwen Yang, Xiaogang Liu, Ligong Wang","doi":"10.13001/ela.2023.7503","DOIUrl":null,"url":null,"abstract":"Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ be the sum of the $k$ largest absolute values of Laplacian eigenvalues of $G$. Let $C_n^+$ be a digraph with $n+1$ vertices obtained from the directed cycle $C_n$ by attaching a pendant arc whose tail is on $C_n$. A digraph is $\\mathbb{C}_n^+$-free if it contains no $C_{\\ell}^+$ as a subdigraph for any $2\\leq \\ell \\leq n-1$. In this paper, we present lower bounds of $S_n(G)$ of digraphs of order $n$. We provide the exact values of $S_k(G)$ of directed cycles and $\\mathbb{C}_n^+$-free unicyclic digraphs. Moreover, we obtain upper bounds of $S_k(G)$ of $\\mathbb{C}_n^+$-free digraphs which have vertex-disjoint directed cycles.","PeriodicalId":50540,"journal":{"name":"Electronic Journal of Linear Algebra","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs\",\"authors\":\"Xiuwen Yang, Xiaogang Liu, Ligong Wang\",\"doi\":\"10.13001/ela.2023.7503\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ be the sum of the $k$ largest absolute values of Laplacian eigenvalues of $G$. Let $C_n^+$ be a digraph with $n+1$ vertices obtained from the directed cycle $C_n$ by attaching a pendant arc whose tail is on $C_n$. A digraph is $\\\\mathbb{C}_n^+$-free if it contains no $C_{\\\\ell}^+$ as a subdigraph for any $2\\\\leq \\\\ell \\\\leq n-1$. In this paper, we present lower bounds of $S_n(G)$ of digraphs of order $n$. We provide the exact values of $S_k(G)$ of directed cycles and $\\\\mathbb{C}_n^+$-free unicyclic digraphs. Moreover, we obtain upper bounds of $S_k(G)$ of $\\\\mathbb{C}_n^+$-free digraphs which have vertex-disjoint directed cycles.\",\"PeriodicalId\":50540,\"journal\":{\"name\":\"Electronic Journal of Linear Algebra\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-07-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Linear Algebra\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.13001/ela.2023.7503\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Linear Algebra","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.13001/ela.2023.7503","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs
Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ be the sum of the $k$ largest absolute values of Laplacian eigenvalues of $G$. Let $C_n^+$ be a digraph with $n+1$ vertices obtained from the directed cycle $C_n$ by attaching a pendant arc whose tail is on $C_n$. A digraph is $\mathbb{C}_n^+$-free if it contains no $C_{\ell}^+$ as a subdigraph for any $2\leq \ell \leq n-1$. In this paper, we present lower bounds of $S_n(G)$ of digraphs of order $n$. We provide the exact values of $S_k(G)$ of directed cycles and $\mathbb{C}_n^+$-free unicyclic digraphs. Moreover, we obtain upper bounds of $S_k(G)$ of $\mathbb{C}_n^+$-free digraphs which have vertex-disjoint directed cycles.
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