On the $A_α$-index of graphs with given order and dissociation number

Abstract

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of $G$. The adjacency matrix and the degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G),$ respectively. In 2017, Nikiforov proposed the $A_α$-matrix: $A_α(G)=αD(G)+(1-α)A(G),$ where $α\in[0,1].$ The largest eigenvalue of this novel matrix is called the $A_α$-index of $G.$ In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest $A_α$-index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the $n$-vertex graphs having the minimum $A_α$-index with dissociation number $τ$, where $τ\geqslant\lceil\frac{2}{3}n\rceil.$ Finally, we identify all the connected $n$-vertex graphs with dissociation number $τ\in\{2,\lceil\frac{2}{3}n\rceil,n-1,n-2\}$ having the minimum $A_α$-index.

Figures (6)

• Figure 1: Tree $S_{n,\tau}^\dag$ together with some labeled vertices.
• Figure 2: Trees $S_{k_1,k_2}, T^1_{r_1,p_1}$ and $T^2_{r_2,p_2}$
• Figure 3: Trees $T_a\in \mathscr{T}^1_{n,\tau}, \ T_b\in\mathscr{T}^2_{n,\tau}$ and $T_c\in\mathscr{T}^3_{n,\tau}$ together with some labeled vertices.
• Figure 4: Trees $T^\dag$ and $T^\dag_2$.
• Figure 5: Trees $T^i_{r_i,p_i}$ for $3\leqslant i\leqslant 8.$

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1: V.N
• Lemma 2.2: C.Y
• Lemma 2.3: C.Y
• Lemma 2.4: HZ
• Lemma 2.5: CS