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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.22015 |
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| _version_ | 1866909711099494400 |
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| author | Kannan, M. Rajesh Roy, Rahul |
| author_facet | Kannan, M. Rajesh Roy, Rahul |
| contents | The algebraic connectivity $a(G)$ of a graph $G$ is defined as the second smallest eigenvalue of its Laplacian matrix $L(G)$. It also admits a variational characterization as the minimum of a quadratic form associated with $L(G)$, subject to $l_2$-norm constraints. In 2024, Andrade and Dahl investigated an analogous parameter $γ(G)$, defined using the $l_\infty$-norm instead of the $l_2$-norm. They demonstrated that $γ(G)$ can be computed in polynomial time using linear programming. In this article, we study the combinatorial significance of $γ(G)$, revealing that it can be efficiently computed using a breadth-first search (BFS) algorithm. We show that $γ(G)$ characterizes the connectedness of the graph $G$. We further establish new bounds on $γ(G)$, and analyze the graphs that attain extremal values. Finally, we derive an elegant formula for $γ(G)$ when $G$ is the Cartesian product of finitely many graphs. Applying this formula, we explicitly compute $γ(G)$ for various families of graphs, including hypercube graphs, Hamming graphs, and others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $l_\infty$-analog of Algebraic Connectivity Kannan, M. Rajesh Roy, Rahul Combinatorics 05C12, 05C50, 05C76 The algebraic connectivity $a(G)$ of a graph $G$ is defined as the second smallest eigenvalue of its Laplacian matrix $L(G)$. It also admits a variational characterization as the minimum of a quadratic form associated with $L(G)$, subject to $l_2$-norm constraints. In 2024, Andrade and Dahl investigated an analogous parameter $γ(G)$, defined using the $l_\infty$-norm instead of the $l_2$-norm. They demonstrated that $γ(G)$ can be computed in polynomial time using linear programming. In this article, we study the combinatorial significance of $γ(G)$, revealing that it can be efficiently computed using a breadth-first search (BFS) algorithm. We show that $γ(G)$ characterizes the connectedness of the graph $G$. We further establish new bounds on $γ(G)$, and analyze the graphs that attain extremal values. Finally, we derive an elegant formula for $γ(G)$ when $G$ is the Cartesian product of finitely many graphs. Applying this formula, we explicitly compute $γ(G)$ for various families of graphs, including hypercube graphs, Hamming graphs, and others. |
| title | On the $l_\infty$-analog of Algebraic Connectivity |
| topic | Combinatorics 05C12, 05C50, 05C76 |
| url | https://arxiv.org/abs/2507.22015 |