| viXra.org > Combinatorics and Graph Theory > viXra:2012.0135 |
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| viXra.org > Combinatorics and Graph Theory > viXra:2012.0135 |
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Combinatorics and Graph Theory |
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Authors: Tariq Alraqad, Akbar Ali, Hicham Saber
Let $G$ be a graph containing no component isomorphic to the path graph of order $2$. Denote by $d_w$ the degree of a vertex $w$ in $G$. The augmented Zagreb index ($AZI$) of $G$ is the sum of the quantities $(d_ud_v/(d_u+d_v-2))^3$ over all edges $uv$ of $G$. Denote by $\mathcal{G}(n,\chi)$ the class of all connected graphs of a fixed order $n$ and with a fixed chromatic number $\chi$, where $n\ge5$ and $3\le \chi \le n-1$. The problem of finding graph(s) attaining the maximum $AZI$ in the class $\mathcal{G}(n,\chi)$ has been solved recently in [F. Li, Q. Ye, H. Broersma, R. Ye, MATCH Commun. Math. Comput. Chem. 85 (2021) 257--274] for the case when $n$ is a multiple of $\chi$. The present paper gives the solution of the aforementioned problem not only for the remaining case (that is, when $n$ is not a multiple of $\chi$) but also for the case considered in the aforesaid paper.
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