New results on $B_α$-eigenvalues of a graph

TL;DR

This work studies the spectral properties of the $B_{\alpha}$-matrix, $B_{\alpha}(G)=\alpha A(G)+(1-\alpha)L(G)$, for graphs $G$ across $\alpha\in[0,1]$ and develops multiplicity results tied to vertex structure (independent sets, cliques, twins) as well as pendant/quasi-pendant configurations. It establishes that $B_{\alpha}(G)$ exhibits nonmonotone $\alpha$-eigenvalues, but $\lambda_1(B_{\\alpha}(G))$ is convex in $\alpha$ and $\lambda_n(B_{\\alpha}(G))$ is concave in $\alpha$, and provides exact multiplicity formulas for $1-\alpha$ in graphs with pendant structures via quotient matrices. The PSD analysis yields a sharp threshold $\beta_0(G)$, with $\beta_0(G)=\frac{2}{3}$ for bipartite graphs and explicit values for the family $\mathcal{H}_n^{\ell}$, linking PSD ranges to graph structure and connecting to the classical $\alpha_0(G)$ problem. Overall, the paper advances the theory of $B_{\alpha}$-matrices by deriving structural multiplicity results, exact spectra for special graph families, and PSD criteria, while identifying open questions for non-bipartite graphs.

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and Laplacian matrix $L(G)$. In 2024, Samanta \textit{et} \textit{al.} defined the convex linear combination of $A(G)$ and $L(G)$ as $B_α(G) = αA(G) + (1-α)L(G)$, for $α\in [0,1]$. This paper presents some results on the eigenvalues of $B_α(G)$ and their multiplicity when some sets of vertices satisfy certain conditions. Moreover, the positive semidefiniteness problem of $B_α(G)$ is studied.

Figures (8)

• Figure 1: An example of the nonmonotonicity of the eigenvalues of $B_{\alpha}(G)$ for $0 \leq \alpha \leq 1$.
• Figure 2: Behavior the $B_{\alpha}$-eigenvalues of $G$ and $\tilde{G}$ for $\alpha \in \left(\dfrac{2}{3}, 1\right]$
• Figure 3: Graph $G$.
• Figure 4: Graph $G$ with $p(G)=9, q(G)= 5$ and $r(G)=9.$
• Figure 5: Relationship between the labeling of the internal vertices and the global labeling.

Theorems & Definitions (52)

• Definition 1: you
• Lemma 1: you
• Proposition 1: Corollary 4.3.15, HoJosecond
• Corollary 1: HoJosecond
• Lemma 2: RM
• Corollary 2: ACPR
• Proposition 2
• Proposition 3
• proof
• Remark 1