Hypergraphs excel as a tool for representing intricate relationships across social, biological, and chemical domains. They offer a robust framework capable of capturing complex forms of interaction that traditional graphs may struggle to depict. Let \(\aleph =(\Lambda ,\Phi )\) be a hypergraph with vertices \(\Lambda =\{\lambda _1,\lambda _2, \dots \lambda _n\}\) and hyperedges \(\Phi =\{\phi _1,\phi _2, \dots \phi _m\}\) . In this context, two hyperedges are considered adjacent if they share at least one vertex. The neighborhood \(N(\lambda _i)\) of a vertex \(\lambda _i\) is defined as the set of vertices that appear in hyperedges adjacent to \(\lambda _i\) . Here, we focus on a specific type of hypergraph known as the 3-uniform hyperstar. Our investigation involves a thorough examination of characteristic equations using adjacency matrices of both 3-uniform hyperstars and general k-uniform hyperstars. We derive recurrence relations for certain coefficients within these characteristic equations to deepen our understanding. The characteristic equation of degree n is expressed as \(-\Omega ^n + C_{n-1}\Omega ^{n-1}+C_{n-2}\Omega ^{n-2}+---+C_3\Omega ^3+C_2\Omega ^2+C_1 \Omega +C_0\) . For a \(\varrho \) -uniform hyperstar, where the number of vertices n is given by \(n=(\mu +1)\varrho -\mu +\varrho -1\) with \(\mu =1,2,3,\dots \) , the sum of the product of two eigenvalues is \(\sum \limits _{i,j=1}^{(\mu +1)\varrho -\mu +\varrho -1} \omega _i \omega _j =\frac {\varrho }{2}[(\mu +2)\varrho -\mu -2] \) or equivalently \(\mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=\frac {\varrho }{2}[(\mu +2)\varrho -\mu -2]\) , where the recurrence relation is \( \mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=\mathcal {P}[(\mu +1)\varrho -\mu ]+\frac {\varrho (\varrho -1)}{2}\) and \(\mathcal {P}(2\varrho -1)=\varrho (\varrho -1) \) . The function \(s(j)=\sum \limits _{i=1}^{j}\frac {i(i+1)}{2}\) is defined to prove the following statement: the sum of the product of three eigen values of a \(\varrho \) -uniform hyperstar is \(\sum \limits _{i,j,k=1}^{(\mu +1)\varrho -\mu +\varrho -1} \omega _i \omega _j \omega _k = s(\varrho -2)\ 2(\mu +2)\) or \(\mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=s(\varrho -2) \ 2(\mu +2)\) , where the recurrence relation is \(\mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=\mathcal {P}[(\mu +1)\varrho -\mu ]+2\ s(\varrho -2)\) and \(\mathcal {P}(2\varrho -1)=4\ s(\varrho -2)\) .

Some Eigen Value Properties of Uniform Hyperstar Using Recurrence Relation

会议论文
发表日期：

2025年

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R. Deepthi Chandran,
P. B. Ramkumar

摘要

Hypergraphs excel as a tool for representing intricate relationships across social, biological, and chemical domains. They offer a robust framework capable of capturing complex forms of interaction that traditional graphs may struggle to depict. Let \(\aleph =(\Lambda ,\Phi )\) be a hypergraph with vertices \(\Lambda =\{\lambda _1,\lambda _2, \dots \lambda _n\}\) and hyperedges \(\Phi =\{\phi _1,\phi _2, \dots \phi _m\}\) . In this context, two hyperedges are considered adjacent if they share at least one vertex. The neighborhood \(N(\lambda _i)\) of a vertex \(\lambda _i\) is defined as the set of vertices that appear in hyperedges adjacent to \(\lambda _i\) . Here, we focus on a specific type of hypergraph known as the 3-uniform hyperstar. Our investigation involves a thorough examination of characteristic equations using adjacency matrices of both 3-uniform hyperstars and general k-uniform hyperstars. We derive recurrence relations for certain coefficients within these characteristic equations to deepen our understanding. The characteristic equation of degree n is expressed as \(-\Omega ^n + C_{n-1}\Omega ^{n-1}+C_{n-2}\Omega ^{n-2}+---+C_3\Omega ^3+C_2\Omega ^2+C_1 \Omega +C_0\) . For a \(\varrho \) -uniform hyperstar, where the number of vertices n is given by \(n=(\mu +1)\varrho -\mu +\varrho -1\) with \(\mu =1,2,3,\dots \) , the sum of the product of two eigenvalues is \(\sum \limits _{i,j=1}^{(\mu +1)\varrho -\mu +\varrho -1} \omega _i \omega _j =\frac {\varrho }{2}[(\mu +2)\varrho -\mu -2] \) or equivalently \(\mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=\frac {\varrho }{2}[(\mu +2)\varrho -\mu -2]\) , where the recurrence relation is \( \mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=\mathcal {P}[(\mu +1)\varrho -\mu ]+\frac {\varrho (\varrho -1)}{2}\) and \(\mathcal {P}(2\varrho -1)=\varrho (\varrho -1) \) . The function \(s(j)=\sum \limits _{i=1}^{j}\frac {i(i+1)}{2}\) is defined to prove the following statement: the sum of the product of three eigen values of a \(\varrho \) -uniform hyperstar is \(\sum \limits _{i,j,k=1}^{(\mu +1)\varrho -\mu +\varrho -1} \omega _i \omega _j \omega _k = s(\varrho -2)\ 2(\mu +2)\) or \(\mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=s(\varrho -2) \ 2(\mu +2)\) , where the recurrence relation is \(\mathcal {P}[(\mu +1)\varrho -\mu +\varrho -1]=\mathcal {P}[(\mu +1)\varrho -\mu ]+2\ s(\varrho -2)\) and \(\mathcal {P}(2\varrho -1)=4\ s(\varrho -2)\) .