On the $A_α$-index of graphs with given order and dissociation number
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913286404964352 |
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| author | Zhou, Zihan Li, Shuchao |
| author_facet | Zhou, Zihan Li, Shuchao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18522 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $A_α$-index of graphs with given order and dissociation number Zhou, Zihan Li, Shuchao Combinatorics 05C50, 15A18 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. |
| title | On the $A_α$-index of graphs with given order and dissociation number |
| topic | Combinatorics 05C50, 15A18 |
| url | https://arxiv.org/abs/2403.18522 |