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On the \(\alpha \)-spectral radius of unicyclic and bicyclic graphs with a fixed diameter

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Abstract

The \(\alpha \)-spectral radius of a connected graph G is the spectral radius of \(A_\alpha \)-matrix of G. In this paper, we discuss the methods for comparing \(\alpha \)-spectral radius of graphs. As applications, we characterize the graphs with the maximal \(\alpha \)-spectral radius among all unicyclic and bicyclic graphs of order n with diameter d, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order n with diameter d. From our conclusion, it is known that the result of Pai and Liu in (Ars Combin 249–265, 2017) is wrong.

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Funding

The funding support for this research was provided by National Natural Science Foundation of China under grant number No. 12271182.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, Shanghai, People’s Republic of China

    Feifei Wang, Haiying Shan & Yuyao Zhai

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  1. Feifei Wang
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  2. Haiying Shan
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  3. Yuyao Zhai
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Correspondence to Haiying Shan.

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A The expressions of \(f_{ij}\) for \(i,j=1,2,3,4\)

A The expressions of \(f_{ij}\) for \(i,j=1,2,3,4\)

Table 2 The expressions of \(f_{ij}\) for \(i,j=1,2,3,4\)
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Wang, F., Shan, H. & Zhai, Y. On the \(\alpha \)-spectral radius of unicyclic and bicyclic graphs with a fixed diameter. Comp. Appl. Math. 42, 144 (2023). https://doi.org/10.1007/s40314-023-02281-2

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  • DOI: https://doi.org/10.1007/s40314-023-02281-2

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