Quadratic embedding constants of fan graphs and graph joins
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916668103458816 |
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| author | Młotkowski, Wojciech Obata, Nobuaki |
| author_facet | Młotkowski, Wojciech Obata, Nobuaki |
| contents | We derive a general formula for the quadratic embedding constant of a graph join $\bar{K}_m+G$, where $\bar{K}_m$ is the empty graph on $m\ge1$ vertices and $G$ is an arbitrary graph. Applying our formula to a fan graph $K_1+P_n$, where $K_1=\bar{K}_1$ is the singleton graph and $P_n$ is the path on $n\ge1$ vertices, we show that $\mathrm{QEC}(K_1+P_n)=-\tildeα_n-2$, where $\tildeα_n$ is the minimal zero of a new polynomial $Φ_n(x)$ related to Chebyshev polynomials of the second kind. Moreover, for an even $n$ we have $\tildeα_n=\min\mathrm{ev}(A_n)$, where the right-hand side is the An minimal eigenvalue of the adjacency matrix $A_n$ of $P_n$. For an odd $n$ we show that $\min\mathrm{ev}(A_{n+1})\le\tildeα_n<\min\mathrm{ev}(A_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16348 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic embedding constants of fan graphs and graph joins Młotkowski, Wojciech Obata, Nobuaki Combinatorics 05C50, 05C12, 05C76, 33C47 We derive a general formula for the quadratic embedding constant of a graph join $\bar{K}_m+G$, where $\bar{K}_m$ is the empty graph on $m\ge1$ vertices and $G$ is an arbitrary graph. Applying our formula to a fan graph $K_1+P_n$, where $K_1=\bar{K}_1$ is the singleton graph and $P_n$ is the path on $n\ge1$ vertices, we show that $\mathrm{QEC}(K_1+P_n)=-\tildeα_n-2$, where $\tildeα_n$ is the minimal zero of a new polynomial $Φ_n(x)$ related to Chebyshev polynomials of the second kind. Moreover, for an even $n$ we have $\tildeα_n=\min\mathrm{ev}(A_n)$, where the right-hand side is the An minimal eigenvalue of the adjacency matrix $A_n$ of $P_n$. For an odd $n$ we show that $\min\mathrm{ev}(A_{n+1})\le\tildeα_n<\min\mathrm{ev}(A_n)$. |
| title | Quadratic embedding constants of fan graphs and graph joins |
| topic | Combinatorics 05C50, 05C12, 05C76, 33C47 |
| url | https://arxiv.org/abs/2403.16348 |