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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.15470 |
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| _version_ | 1866913241601409024 |
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| author | Junior, Joao Domingos Gomes da Silva Oliveira, Carla Silva da Costa, Liliana Manuela Gaspar C. |
| author_facet | Junior, Joao Domingos Gomes da Silva Oliveira, Carla Silva da Costa, Liliana Manuela Gaspar C. |
| contents | Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_α$-matrix of $G$, $A_α(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_α(G):=αA(G)+(1-α)D(G),$ where $α\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_α(G)$ and for the largest and smallest eigenvalues of $A_α(l(G))$. Extremal graphs attaining some of these bounds are characterized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some results involving the $A_α$-eigenvalues for graphs and line graphs Junior, Joao Domingos Gomes da Silva Oliveira, Carla Silva da Costa, Liliana Manuela Gaspar C. Discrete Mathematics Combinatorics 05C05 Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_α$-matrix of $G$, $A_α(G)$, as a linear convex combination of $A(G)$ and $D(G)$, the following way, $A_α(G):=αA(G)+(1-α)D(G),$ where $α\in[0,1]$. In this paper, we present some bounds for the eigenvalues of $A_α(G)$ and for the largest and smallest eigenvalues of $A_α(l(G))$. Extremal graphs attaining some of these bounds are characterized. |
| title | Some results involving the $A_α$-eigenvalues for graphs and line graphs |
| topic | Discrete Mathematics Combinatorics 05C05 |
| url | https://arxiv.org/abs/2402.15470 |