Structural and extremal properties of $l_1$-Fiedler value

TL;DR

This paper explores the properties and extremal values of the $l_1$-Fiedler value, a graph parameter based on the $ ext{l}_1$-norm, including bounds, extremal structures, and relationships with graph connectivity and isoperimetric properties.

Contribution

It introduces and analyzes the $l_1$-Fiedler value, providing bounds, extremal graphs, and connections to Laplacian matrices and graph invariants, expanding understanding of algebraic connectivity variants.

Findings

Derived a Nordhaus--Gaddum type inequality for $b(G)$.

Identified extremal trees with respect to $b(G)$, including maximizers and minimizers.

Established bounds for $b(G)$ based on edge connectivity and characterized graphs attaining these bounds.

Abstract

The algebraic connectivity $a(G)$, defined as the second smallest eigenvalue of the Laplacian matrix $L(G)$, admits a well-known variational characterization involving the minimization of a quadratic form subject to an $\ell_{2}$-norm constraint. In a recent work, Andrade and Dahl (2024) proposed an analogous formulation based on the $\ell_{1}$-norm, leading to the introduction of a new graph parameter $b(G)$, referred to as the $l_1$-Fiedler value. In this article, we undertake a detailed investigation of the structural and extremal properties of $b(G)$. We first derive a Nordhaus--Gaddum type inequality for $b(G)$. For trees, we determine both global maximizer and minimizers of $b(G)$, and present extremal constructions for trees with prescribed diameter, maximum degree, and number of pendant vertices. We further establish a connection between $b(G)$ and Laplacian matrices, and obtain a bound for $b(G)$ in terms of the edge connectivity, along with a complete characterization of the graphs attaining equality. We derive an explicit formula that describes the behaviour of $b(G)$ under the addition of pendant vertices. We also investigate the connection between $b(G)$ and the isoperimetric number.