Abstract
In this article, we present an enhanced version of the symmetric division deg index (sdd-index) known as symmetric division eccentric index or SDE index, for short. Unlike its predecessor, SDE employs eccentricity instead of vertex degree to assess the properties of a graph G. In this paper, we first give some bounds for SDE index of a connected graph G with fixed size m. For two connected graphs \({\varvec{G}}_{\varvec{1}}\) and \({\varvec{G}}_{\varvec{2}}\) of order \({\varvec{n}}_{\varvec{1}}\) and \({\varvec{n}}_{\varvec{2}}\), employing these bounds, we compute the SDE index for two classes of graph products, e.g., the Cartesian product and Corona product. As an application, we determine the structure of graphs with two non-equi-centric edges. Our theorems generalize the recent results for the extended adjacency index of a graph. Besides, this research significantly contributes to the comprehension of graph analysis techniques and offers valuable insights into the relationship between SDE and various graph properties.
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References
Buckley, F., Harary, F.: Distance in Graphs. Addison-Wesley, Redwood (1990)
Chen, J., Zheng, S., Zhao, H., Yang, Y.: Structure-aware protein solubility prediction from sequence through graph convolutional network and predicted contact map. J. Cheminform. 13(1), 1–10 (2021)
Estrada, E., Torres, L., Rodriguez, L., Gutman, I.: An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. Sect. 37A, 849–855 (1998)
Fajtlowicz, S.: On conjectures of Graffiti-II. Congr. Numer. 60, 187–197 (1987)
Frucht, R., Harary, F.: On the corona of two graphs. Aequ. Math. 4(3), 322–325 (1970)
Ghorbani, M., Alidehi-Ravandi, R., Dehmer, M.: Fullerenes via their counting polynomials. Appl. Math. Comput. 466, 128431 (2024)
Ghorbani, M., Alidehi-Ravandi, R., Dehmer, M., Emmert-Streib, F.: A study of roots of a certain class of counting polynomials. Mathematics 11, 2876 (2023)
Gupta, C.K., Lokesha, V., Shwetha Shetty, B., Ranjini, P.S.: Graph operations on the symmetric division deg index of graphs. Palestine J. Math 5, 1–18 (2016)
Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17(4), 535–538 (1972)
Huang, Y.-A., Hu, P., Chan, K.C., You, Z.-H.: Graph convolution for predicting associations between miRNA and drug resistance. Bioinformatics 36, 851–858 (2020)
Jha, K., Saha, S., Singh, H.: Prediction of protein-protein interaction using graph neural networks. Sci. Rep. 12(1), 60–83 (2022)
Randić, M.: On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975)
Vukičević, D.: Bond additive modeling 2 mathematical properties of max-min rodeg index. Croat. Chem. Acta 83(3), 261–273 (2010)
West, D.B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (2001)
Zhou, J., Cui, G., Hu, S., Zhang, Z., Yang, C., Liu, Z., Wang, L., Li, C., Sun, M.: Graph neural networks: a review of methods and applications. AI Open 1, 57–81 (2020)
Zhou, B., Trinajstić, N.: On a novel connectivity index. J. Math. Chem. 46, 1252–1270 (2009)
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This research is partially supported by Shahid Rajaee Teacher Training University under Grant Number 5036.
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Ghorbani, M., Alidehi-Ravandi, R. Exploring the SDE index: a novel approach using eccentricity in graph analysis. J. Appl. Math. Comput. 70, 947–967 (2024). https://doi.org/10.1007/s12190-023-01980-7
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DOI: https://doi.org/10.1007/s12190-023-01980-7