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2022•arXiv (Cornell University)Open access

A New Upper Bound for the d-dimensional Algebraic Connectivity of Arbitrary Graphs

Juan F. Presenza, Ignacio Mas, Juan I. Giribet, J. Ignacio Alvarez-Hamelin

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Abstract

In this paper we show that the $d$-dimensional algebraic connectivity of an arbitrary graph $G$ is bounded above by its $1$-dimensional algebraic connectivity, i.e., $a_d(G) \leq a_1(G)$, where $a_1(G)$ corresponds the well-studied second smallest eigenvalue of the graph Laplacian.

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In this paper we show that the $d$-dimensional algebraic connectivity of an arbitrary graph $G$ is bounded above by its $1$-dimensional algebraic connectivity, i.e., $a_d(G) \leq a_1(G)$, where $a_1(G)$ corresponds the well-studied second smallest eigenvalue of the graph Laplacian.

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Available abstract

In this paper we show that the $d$-dimensional algebraic connectivity of an arbitrary graph $G$ is bounded above by its $1$-dimensional algebraic connectivity, i.e., $a_d(G) \leq a_1(G)$, where $a_1(G)$ corresponds the well-studied second smallest eigenvalue of the graph Laplacian.

Key concepts: Algebraic connectivity, Algebraic number, Bounded function, Laplacian matrix, Mathematics, Algebraic graph theory, Eigenvalues and eigenvectors, Graph

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