Characterizing $A_\alpha$-minimizer graphs: given order and independence number

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