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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 1

Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices

Jia Bao Liu
Shaohui Wang
Chunxiang Wang
Sayfa
33–40
Yayın Tarihi
2022-10-18
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

As consequences, we determine the graph with the $A_{\alpha}$-spectral radius among all the cacti with $n$ vertices; we also characterize the $n$-vertex cacti with a perfect matching having the largest $A_{\alpha}$-spectral radius.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:For $ \alpha \in [0,1]$, let $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ be $A_{\alpha}$-matrix, where $A(G)$ is the adjacent matrix and $D(G)$ is the diagonal matrix of the degrees of a graph $G$.
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Temel Bulgular & Veri:Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\frac{1}{2}}$ is the signless Laplacian matrix.
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Sonuç & Katkı:As consequences, we determine the graph with the $A_{\alpha}$-spectral radius among all the cacti with $n$ vertices; we also characterize the $n$-vertex cacti with a perfect matching having the largest $A_{\alpha}$-spectral radius.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Adjacent matrix · trees · cacti

Abstract

For $ \alpha \in [0,1]$, let $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ be $A_{\alpha}$-matrix, where $A(G)$ is the adjacent matrix and $D(G)$ is the diagonal matrix of the degrees of a graph $G$. Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\frac{1}{2}}$ is the signless Laplacian matrix. A connected graph is a cactus graph if any two cycles of $G$ have at most one common vertex. We first propose the result for subdivision graphs, and determine the cacti maximizing $A_{\alpha}$-spectral radius subject to fixed pendant vertices. In addition, the corresponding extremal graphs are provided. As consequences, we determine the graph with the $A_{\alpha}$-spectral radius among all the cacti with $n$ vertices; we also characterize the $n$-vertex cacti with a perfect matching having the largest $A_{\alpha}$-spectral radius.

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Bu Makaleye Atıf Yap

Liu, J. B., Wang, S., Wang, C. (2022). Sharp upper bounds of $A_\alpha$-spectral radius of cacti with given pendant vertices. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 1, 33–40. https://ajindex.com/articles/529476

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