[Paper Review] On the $\alpha$-spectral radius of graphs

Haiyan Guo, Bo Zhou|arXiv (Cornell University)|May 9, 2018

Graph theory and applications13 references3 citations

TL;DR

This paper investigates the α-spectral radius ρα(G), the largest eigenvalue of Aα(G) = αD(G) + (1−α)A(G), for various graph classes. It establishes tight upper bounds for unicyclic and irregular graphs, identifies the unique trees with second-maximal and diameter-constrained maximal α-spectral radius, and proves edge-relocation effects on the spectral radius, culminating in the characterization of graphs maximizing the difference between maximum degree and ρα(G).

ABSTRACT

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $\alpha$-spectral radius of $G$ is the largest eigenvalue of $A_{\alpha}(G)$. We give upper bounds for $\alpha$-spectral radius for unicyclic graphs $G$ with maximum degree $\Delta\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $\alpha$-spectral radius among trees, and the unique tree with maximum $\alpha$-spectral radius among trees with given diameter. For a graph with two pendant paths at a vertex or at two adjacent vertex, we prove results concerning the behavior of the $\alpha$-spectral radius under relocation of a pendant edge in a pendant path. We also determine the unique graphs such that the difference between the maximum degree and the $\alpha$-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.

Motivation & Objective

• To extend known upper bounds for the α-spectral radius from trees to unicyclic graphs.
• To determine the unique tree with second-largest α-spectral radius among all trees.
• To identify the unique tree with maximum α-spectral radius among trees with a given diameter.
• To analyze how relocating a pendant edge in a pendant path affects the α-spectral radius.
• To characterize the unique graphs that maximize the difference between maximum degree and α-spectral radius in trees, unicyclic graphs, and non-bipartite graphs.

Proposed method

• Uses the matrix Aα(G) = αD(G) + (1−α)A(G), where D(G) is the degree matrix and A(G) the adjacency matrix.
• Applies the Perron-Frobenius theorem to ensure a unique positive eigenvector (Perron vector) for connected graphs when 0 ≤ α < 1.
• Employs variational methods and eigenvalue interlacing to compare ρα(G) and ρα(G′) under graph modifications.
• Leverages Lemma 2.1 and Corollary 2.1 to prove that certain edge-relocation operations strictly increase ρα(G).
• Uses algebraic manipulation and root analysis of cubic polynomials to compute ρα(Sn + e), the α-spectral radius of the star with one extra edge.
• Applies the Cauchy-Schwarz inequality and trace identities to derive bounds on spectral energy and Estrada index.

Experimental results

Research questions

• RQ1 What is the tightest upper bound for the α-spectral radius of unicyclic graphs with maximum degree ∆ ≥ 2?
• RQ2 Which tree has the second-largest α-spectral radius among all trees of order n?
• RQ3 Which tree has the largest α-spectral radius among all trees with a fixed diameter?
• RQ4 How does relocating a pendant edge in a pendant path affect the α-spectral radius, and can this be proven rigorously?
• RQ5 Which graphs maximize the difference between the maximum degree and the α-spectral radius in the classes of trees, unicyclic graphs, and non-bipartite graphs?

Key findings

• The upper bound for the α-spectral radius of trees with maximum degree ∆ ≥ 2 also holds for unicyclic graphs.
• The unique tree with the second-largest α-spectral radius among all trees is the double star S_{2,n-2} with a central edge connecting vertices of degrees 2 and n−2.
• The unique tree with maximum α-spectral radius among trees of given diameter d is the double star S_{k,d−k} for appropriate k.
• For graphs with two pendant paths at a vertex or two adjacent vertices, relocating a pendant edge toward the center increases the α-spectral radius, confirming a conjecture from [26].
• The unique graph maximizing the difference between maximum degree and α-spectral radius is the star with one additional edge, Sn + e, in the classes of trees, unicyclic graphs, and non-bipartite graphs.
• The α-spectral radius of Sn + e is the largest root of the cubic equation h(t) = t³ − (α(n+1)+1)t² + ((α²+3α−1)(n−1)+α(α+1))t + (1−2α)(α+1)(n−1) − 2(1−α)² = 0.