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Corrections and Further Spectral Results for the Graovac–Ghorbani Matrix | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 11 آبان 1405 | ||
| نوع مقاله: Erratum | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.32029.3057 | ||
| نویسنده | ||
| Murat Cancan* | ||
| Faculty of Education, Van Yuzuncu Yil University Van, Turkey | ||
| چکیده | ||
| The Graovac--Ghorbani index of a connected graph $G$ is $\mathcal A(G)=\sum_{uv\in E(G)}\sqrt{\frac{n_u+n_v-2}{n_u,n_v}},$ where $n_u$ is the number of vertices closer to $u$ than to $v$. Some statements in the paper of Alidehi-Ravandi and Ghorbani \emph{(Commun. Comb. Optim., 2026)} contain several inaccuracies. We identify these inaccuracies, provide connected counterexamples, and give corrected statements. The corrections are the main part of the note. We also give an entrywise form of the bipartite bound $\rho(\widetilde A)\le\sqrt{n-2}$, which is already Theorem~9 of the original paper, a Randić-type bound for the index that follows from it, and a lower bound for the spectral radius of bipartite graphs, checked against the classical bounds on examples. | ||
| کلیدواژهها | ||
| Graovac–Ghorbani matrix؛ spectral radius؛ counterexample؛ strongly distance- balanced graph | ||
| مراجع | ||
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[1] R. Alidehi-Ravandi and M. Ghorbani, Spectral properties of a new distance-based matrix, Commun. Comb. Optim. (2026), In press. https://doi.org/10.22049/cco.2026.31172.2767
[2] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012.
[3] D. Dimitrov, B. Ikica, and R. Škrekovski, Remarks on the Graovac–Ghorbani index of bipartite graphs, Appl. Math. Comput. 293 (2017), 370–376. https://doi.org/10.1016/j.amc.2016.08.047
[4] A. Graovac and M. Ghorbani, A new version of atom-bond connectivity index, Acta Chim. Slov. 57 (2010), no. 3, 609–612. [5] R.A. Horn and C.R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, 2013.
[6] K. Kutnar, A. Malniˇc, D. Marušič, and Š. Miklavič, Distance-balanced graphs: Symmetry conditions, Discrete Math. 306 (2006), no. 16, 1881–1894. https://doi.org/10.1016/j.disc.2006.03.066
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آمار تعداد مشاهده مقاله: 6 |
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