[Paper Review] The Algebraic Connectivity and the Clique Number of Graphs

Ya-Lei Jin, Xiao‐Dong Zhang|arXiv (Cornell University)|Jun 30, 2013

Graph theory and applications13 references3 citations

TL;DR

This paper establishes the first spectral analog of the Erdős-Stone theorem using algebraic connectivity, characterizing extremal graphs that maximize or minimize algebraic connectivity among graphs of order $n$ with clique number $r$. It proves that the algebraic connectivity of such graphs asymptotically approaches $n(1 - 1/\psi(\mathcal{H}))$, where $\psi(\mathcal{H})$ is the minimum chromatic number minus one over a family $\mathcal{H}$, and identifies Turán graphs and kite graphs as extremal structures for upper and lower bounds, respectively.

ABSTRACT

This paper investigates some relationship between the algebraic connectivity and the clique number of graphs. We characterize all extremal graphs which have the maximum and minimum the algebraic connectivity among all graphs of order $n$ with the clique number $r$, respectively. In turn, an upper and lower bounds for the clique number of a graph in terms of the algebraic connectivity are obtained. Moreover, a spectral version of the Erdős-Stone theorem in terms of the algebraic connectivity of graphs is presented.

Motivation & Objective

• To determine the extremal graphs that achieve maximum and minimum algebraic connectivity among all graphs of order $n$ with clique number $r$.
• To derive upper and lower bounds for the clique number of a graph in terms of its algebraic connectivity.
• To establish a spectral version of the Erdős-Stone theorem using algebraic connectivity as the spectral parameter.
• To characterize the structure of graphs achieving extremal algebraic connectivity values under clique number constraints.

Proposed method

• Use of the Laplacian matrix and its second smallest eigenvalue (algebraic connectivity $\alpha(G)$) as the primary spectral measure.
• Application of graph join operations ($G_1 \vee G_2$) to construct extremal graphs and analyze connectivity properties.
• Employment of iterative edge-deletion techniques to reduce graphs while preserving connectivity and monotonicity of algebraic connectivity.
• Leverage known extremal graph theory results, particularly Turán graphs $T_{n,r}$, as benchmarks for extremal algebraic connectivity.
• Use of spectral graph theory lemmas (e.g., Lemma 4.1, Lemma 4.2, Corollary 4.6) to compare algebraic connectivity across graph families.
• Proof by case analysis on vertex adjacency patterns between the clique and pendant components to show optimality of kite graphs for minimum algebraic connectivity.

Experimental results

Research questions

• RQ1 What is the maximum algebraic connectivity among all graphs of order $n$ with clique number $r$?
• RQ2 What is the minimum algebraic connectivity among all connected graphs of order $n$ with clique number $r$?
• RQ3 How does the algebraic connectivity relate to the clique number in general graphs?
• RQ4 Can the Erdős-Stone theorem be reformulated in spectral terms using algebraic connectivity?
• RQ5 What are the structural characteristics of graphs that extremize algebraic connectivity under clique number constraints?

Key findings

• The maximum algebraic connectivity among graphs of order $n$ with clique number $r$ is achieved by the Turán graph $T_{n,r}$, with $\alpha(G) = n - \lceil n/r \rceil$ when $n = kr$ or $n = kr + r - 1$, and by a join structure involving $t$ isolated vertices and a $K_r$-free graph otherwise.
• The minimum algebraic connectivity among connected graphs of order $n$ with clique number $r$ is achieved by the kite graph $Ki_{n,r}$, which consists of a $K_r$ with a pendant path of length $n - r$ attached to one vertex.
• An asymptotic spectral version of the Erdős-Stone theorem is established: $\lim_{n \to \infty} \frac{\alpha(n, \mathcal{H})}{n} = 1 - \frac{1}{\psi(\mathcal{H})}$, where $\psi(\mathcal{H}) = \min\{\chi(H) \mid H \in \mathcal{H}\} - 1$, linking algebraic connectivity to extremal graph theory.
• An upper bound on the clique number $r$ is derived: $r \leq n + 1 - \frac{4}{n\alpha(G)}$, based on the lower bound $\alpha(G) \geq \frac{4}{n(n - r + 1)}$ for kite graphs.
• The algebraic connectivity of a graph $G$ with clique number $r$ satisfies $\frac{n}{n - \alpha(G)} \leq r$, providing a lower bound on $r$ in terms of $\alpha(G)$.
• For graphs with $n = kr + t$ and $0 < t < r - 1$, equality in the upper bound $\alpha(G) \leq n - \lceil n/r \rceil$ holds only for specific join structures involving $t$ isolated vertices and a $K_{r+1-t}$-free graph, confirming the extremality of Turán-type constructions.