Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913274663010304 |
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| author | Shen, Chenghao Shan, Haiying |
| author_facet | Shen, Chenghao Shan, Haiying |
| contents | The $f$ adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge $ij$ is $f(d_i,d_j)$, where $f$ is a real symmetric function and $d_i,d_j$ are the degrees of vertex $i$ and vertex $j$. The $f$-spectral radius of a graph is the spectral radius of its $f$-adjacency matrix. In this paper, the effect of subdividing an edge on $f$-spectral radius is discussed. Some necessary conditions of the extremal graph with given order and size are derived. As an example, we obtain the bicyclic graph(s) with the smallest $f$-spectral radius for fixed order $n\geq8$ by applying generalized Lu-Man method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13525 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size Shen, Chenghao Shan, Haiying Combinatorics The $f$ adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge $ij$ is $f(d_i,d_j)$, where $f$ is a real symmetric function and $d_i,d_j$ are the degrees of vertex $i$ and vertex $j$. The $f$-spectral radius of a graph is the spectral radius of its $f$-adjacency matrix. In this paper, the effect of subdividing an edge on $f$-spectral radius is discussed. Some necessary conditions of the extremal graph with given order and size are derived. As an example, we obtain the bicyclic graph(s) with the smallest $f$-spectral radius for fixed order $n\geq8$ by applying generalized Lu-Man method. |
| title | Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.13525 |