Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.16015

Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces

J. Ignacio Alvarez-Hamelin·Juan I. Giribet·Ignacio Mas·J. Francisco Presenza
View original source ↗Read source document ↗Export metadata ↓

Abstract

This article presents new contributions to the study of graph rigidity and its interplay with fundamental graph invariants. Recently, a quantitative measure of graph rigidity in $\mathbb{R}^d$, termed the generalized algebraic connectivity, was introduced. This development extends the notion of algebraic connectivity---the second-smallest eigenvalue of the Laplacian matrix---which is commonly used to quantify graph connectivity. In this work, we show that the generalized algebraic connectivity is bounded above by the algebraic connectivity. To capture this relationship, we introduce the $d$-rigidity ratio, a normalized metric of a graph's rigidity relative to its connectivity. We also investigate the relationship between rigidity and the diameter---a measure of the graph's overall extent. In this context, we provide the maximal diameter achievable by rigid graphs and show that generalized path graphs serve as extremal examples. Moreover, we establish a new upper bound for the algebraic connectivity that depends inversely on the diameter and the vertex connectivity, improving upon previous bounds. Finally, we derive an upper bound for the algebraic connectivity of generalized path graphs that asymptotically improves existing ones by a factor of four.

Explore related subjects

math.COmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

J. Ignacio Alvarez-Hamelin, Juan I. Giribet, Ignacio Mas, J. Francisco Presenza. 2026-09-04. Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces. https://doi.org/10.1016/j.laa.2025.07.028

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗
arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

math.CO↗