The algebraic connectivity π β‘ ( πΊ ) , defined as the second smallest eigenvalue of the Laplacian matrix πΏ β‘ ( πΊ ) , admits a well-known variational characterization involving the minimization of a quadratic form subject to an β 2 -norm constraint. In a recent work, Andrade and Dahl (2024) proposed an analogous formulation based on the β 1 -norm, leading to the introduction of a new graph parameter π β‘ ( πΊ ) , referred to as the π 1 -Fiedler value. In this article, we undertake a detailed investigation of the structural and extremal properties of π β‘ ( πΊ ) . We first derive a NordhausβGaddum type inequality for π β‘ ( πΊ ) . For trees, we determine both global maximizer and minimizers of π β‘ ( πΊ ) , and present extremal constructions for trees with prescribed diameter, maximum degree, and number of pendant vertices. We further establish a connection between π β‘ ( πΊ ) and Laplacian matrices, and obtain a bound for π β‘ ( πΊ ) in terms of the edge connectivity, along with a complete characterization of the graphs attaining equality. We derive an explicit formula that describes the behavior of π β‘ ( πΊ ) under the addition of pendant vertices. We also investigate the connection between π β‘ ( πΊ ) and the isoperimetric number.
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Kannan et al. (2026) studied this question.
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