Abstract
The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. In this paper, we give the sharp upper bound on the total eccentricity index over graphs with fixed number of pendant vertices and the sharp lower bound on the same over graphs with fixed number of cut vertices. We also provide the sharp upper bounds on the total eccentricity index over graphs with s cut vertices for \(s=0, 1, n-3, n-2\) and propose a conjecture for \(2\le s\le n-4\).
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References
Buckley, F., Harary, F.: Distance in Graphs. Addison-Wesley, Redwood City (1990)
Cavaleri, M., Donno, A.: Some degree and distance-based invariants of wreath products of graphs. Discrete Appl. Math. 277, 22–43 (2020)
Darabi, H., Alizadeh, Y., Klavžar, S., Das, K.C.: On the relation between Wiener index and eccentricity of a graph. J. Comb. Optim. 41(4), 817–829 (2021)
De, N., Nayeem, A., Md, S., Pal, A.: Total eccentricity index of the generalized hierarchical product of graphs. Int. J. Appl. Comput. Math. 1(3), 503–511 (2015)
Dankelmann, P., Goddard, W., Swart, C.S.: The average eccentricity of a graph and its subgraphs. Util. Math. 65, 41–51 (2004)
Dankelmann, P., Mukwembi, S.: Upper bounds on the average eccentricity. Discrete Appl. Math. 167, 72–79 (2014)
Dankelmann, P., Osaye, F.J.: Average eccentricity, minimum degree and maximum degree in graphs. J. Comb. Optim. 40(3), 697–712 (2020)
Eliasi, M., Taeri, B.: Four new sums of graphs and their Wiener indices. Discrete Appl. Math. 157(4), 794–803 (2009)
Eliasi, M., Raeisi, G., Taeri, B.: Wiener index of some graph operations. Discrete Appl. Math. 160(9), 1333–1344 (2012)
Fathalikhani, K., Faramarzi, H., Yousefi-Azari, H.: Total eccentricity of some graph operations. Electron. Notes Discrete Math. 45, 125–131 (2014)
Ilić, A.: On the extremal properties of the average eccentricity. Comput. Math. Appl. 64(9), 2877–2885 (2012)
Pandey, D., Patra, K.L.: Wiener index of graphs with fixed number of pendant or cut vertices. Czechoslovac Math. J. 72(147), no. 2, 411–431 (2022)
Smith, H., Székely, L., Wang, H.: Eccentricity sums in trees. Discrete Appl. Math. 207, 120–131 (2016)
Tang, Y., West, D.B.: Lower bounds for eccentricity-based parameters of graphs. https://faculty.math.illinois.edu/~west/pubs/avgecc.pdf
Tang, Y., Zhou, B.: Ordering unicyclic graphs with large average eccentricities. Filomat 28(1), 207–210 (2014)
Tang, Y., Zhou, B.: On average eccentricity. MATCH Commun. Math. Comput. Chem. 67(2), 405–423 (2012)
Yu, G., Feng, L., Wang, D.: On the average eccentricity of unicyclic graphs. Ars Combin. 103, 531–537 (2012)
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The authors wish to sincerely thank the referee for his/her valuable comments and suggestions, thus improving the submitted version of the article.
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Dinesh Pandey was supported by UGC Fellowship scheme (Sr. No. 2061641145), Government of India.
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Pandey, D., Patra, K.L. Total eccentricity index of graphs with fixed number of pendant or cut vertices. Ricerche mat (2023). https://doi.org/10.1007/s11587-022-00756-8
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DOI: https://doi.org/10.1007/s11587-022-00756-8