Maximizing the index of signed complete graphs with spanning trees on $k$ pendant vertices
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| Format: | Preprint |
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2024
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| _version_ | 1866909242217201664 |
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| author | Li, Dan Yan, Minghui Teng, Zhaolin |
| author_facet | Li, Dan Yan, Minghui Teng, Zhaolin |
| contents | A signed graph $Σ=(G,σ)$ consists of an underlying graph $G=(V,E)$ with a sign function $σ:E\rightarrow\{-1,1\}$. Let $A(Σ)$ be the adjacency matrix of $Σ$ and $λ_1(Σ)$ denote the largest eigenvalue (index) of $Σ$.Define $(K_n,H^-)$ as a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we focus on the following problem: which spanning tree $T$ with a given number of pendant vertices makes the $λ_1(A(Σ))$ of the unbalanced $(K_n,T^-)$ as large as possible? To answer the problem, we characterize the extremal signed graph with maximum $λ_1(A(Σ))$ among graphs of type $(K_n,T^-)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_11214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximizing the index of signed complete graphs with spanning trees on $k$ pendant vertices Li, Dan Yan, Minghui Teng, Zhaolin Combinatorics 05C35, 05C50 A signed graph $Σ=(G,σ)$ consists of an underlying graph $G=(V,E)$ with a sign function $σ:E\rightarrow\{-1,1\}$. Let $A(Σ)$ be the adjacency matrix of $Σ$ and $λ_1(Σ)$ denote the largest eigenvalue (index) of $Σ$.Define $(K_n,H^-)$ as a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we focus on the following problem: which spanning tree $T$ with a given number of pendant vertices makes the $λ_1(A(Σ))$ of the unbalanced $(K_n,T^-)$ as large as possible? To answer the problem, we characterize the extremal signed graph with maximum $λ_1(A(Σ))$ among graphs of type $(K_n,T^-)$. |
| title | Maximizing the index of signed complete graphs with spanning trees on $k$ pendant vertices |
| topic | Combinatorics 05C35, 05C50 |
| url | https://arxiv.org/abs/2405.11214 |