Structural and extremal properties of $l_1$-Fiedler value
M. Rajesh Kannan, Rahul Roy
TL;DR
The paper investigates the $l_1$-Fiedler value $b(G)$, the $ ext{l}_1$-norm analogue of algebraic connectivity, by formulating it as a sparsest-cut based optimization and relating it to the Laplacian. It provides tight bounds and extremal characterizations: a Nordhaus–Gaddum-type bound, complete determination of extremal trees, and explicit diameter/degree/pendant-vertex extremizers. A key contribution is the explicit linkage between $b(G)$ and Laplacian properties, including a vector $x$ that yields $b(G)$ via $L(G)x$, and a bound in terms of edge-connectivity with a full equality condition. The work further connects $b(G)$ to the isoperimetric number, deriving both upper and lower bounds and constructing cases where $b(G)$ matches $\mathrm{iso}(G)$, thereby highlighting the structural implications of the $ ext{l}_1$-Fiedler value for graph connectivity and sparsest cuts.
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.