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Characterizing $A_α$-minimizer graphs: given order and independence number

Jiaqi Zhang, Shuchao Li

Abstract

For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_α(G) \) matrix, proposed by Nikiforov, is defined as \( A_α(G)=αD(G)+(1 - α)A(G) \), where \( α\in[0,1] \). This matrix captures the gradual transition from \( A(G) \) to \( Q(G) \). Let \( \mathcal{G}_{n,γ} \) denote the family of all connected graphs with \( n \) vertices and independence number \( γ\). A graph in \( \mathcal{G}_{n,γ} \) is referred to as an \( A_α\)-minimizer graph if it achieves the minimum \( A_α\) spectral radius. In this paper, we first demonstrate that the \( A_α\)-minimizer graph in \( \mathcal{G}_{n,γ} \) must be a tree when \( γ\geq\left\lceil\frac{n}{2}\right\rceil \), and we provide several characterizations of such \( A_α\)-minimizer graphs. We then specifically characterize the \( A_α\)-minimizer graphs for the case \( γ= \left\lceil\frac{n}{2}\right\rceil + 1 \) when $n\geq 9$. Furthermore, we obtain a structural characterization for the \( A_α\)-minimizer graph when \( γ=n - c \), where \( c\geq4 \) is an integer.

Characterizing $A_α$-minimizer graphs: given order and independence number

Abstract

For a given graph , let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of , respectively. The \( A_α(G) \) matrix, proposed by Nikiforov, is defined as \( A_α(G)=αD(G)+(1 - α)A(G) \), where . This matrix captures the gradual transition from \( A(G) \) to \( Q(G) \). Let denote the family of all connected graphs with vertices and independence number . A graph in is referred to as an -minimizer graph if it achieves the minimum spectral radius. In this paper, we first demonstrate that the -minimizer graph in must be a tree when , and we provide several characterizations of such -minimizer graphs. We then specifically characterize the -minimizer graphs for the case when . Furthermore, we obtain a structural characterization for the -minimizer graph when , where is an integer.

Paper Structure

This paper contains 7 sections, 42 theorems, 58 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Some known lemmas
  3. Proofs of Theorems \ref{['thm3.01']} and \ref{['thm3.02']}
  4. Proof of Theorem \ref{['thm4.1']}
  5. Some structure characterization on $\hat{T}$
  6. Proof of Theorem \ref{['thm4.1']}
  7. Concluding remarks

Key Result

Theorem 1.1

Let $\gamma \geq \lceil \frac{n}{2} \rceil$ and $\alpha \in [0,1)$. If $G^*$ is an $A_{\alpha}$-minimizer in $\mathcal{G}_{n,\gamma}$, then $G^*$ is a tree.

Figures (5)

  • Figure 1: Graphs $W_n$ and $D_n$.
  • Figure 2: (a) $S(K_{1,3})\circ_{V(K_{1,3})}(\ell(w_1),\ell(w_2),\ell(w_3),\ell(w_4))$; (b) $S(P_4)\circ_{V(P_4)}(\ell(u_1),\ell(u_2),\ell(u_3),\ell(u_4))$.
  • Figure 3: Two graphs $\hat{T},\,T"(H_1)$ in the proofs of Theorems \ref{['thm3.01']} and \ref{['thm3.02']}.
  • Figure 4: Graphs $T^\diamond_i$ and $T^\diamond_{i+1}$
  • Figure 5: Graphs $G_1, G_2, F_{s,t}$ used in the proof of Lemma \ref{['lem5.9']} and Theorem \ref{['thm1.5']}.

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: 1
  • Lemma 2.2: 2
  • Lemma 2.3: 10
  • Lemma 2.4: 4
  • Lemma 2.5: 5
  • Lemma 2.6: 12
  • Lemma 2.7: 6
  • ...and 55 more