Published in Volume XXXV, Issue 2, 2025, pages 197–210, doi:10.47743/SACS.2025.2.197
Author: P. N. Simha, B. P. Shylaja, N. Narahari, H. M. Nagesh, U. V. C. Kumar
Abstract
The energy of a vertex in a graph plays a very important role in terms of its contribution to the total energy of a graph, a significant graph invariant in the field of chemical graph theory. Particularly, the computation of vertex energies of various types of graphs in general and integral graphs has been carried out in recent studies. Working in this direction, we determine the vertex energy of all connected non-regular non-bipartite integral graphs with maximum vertex degree four in this article.
References
[1] Octavio Arizmendi, Jorge Fernandez Hidalgo, and Oliver Juarez-Romero. Energy of a vertex. Linear Algebra and its Applications, 557:464–495, 2018. doi:10.1016/j.laa.2018.08.014.
[2] Octavio Arizmendi and Oliver Juárez Romero. On bounds for the energy of graphs and digraphs. In Fernando Galaz-García, Juan Carlos Pardo Millán, and Pedro Solórzano, editors, Contributions of Mexican Mathematicians Abroad in Pure and Applied Mathematics, volume 709, pages 1–19. American Mathematical Society, 2018. doi:10.1090/conm/709/14288.
[3] Krystyna T. Balińska, Dragŏs Cvetković, Zoran Radosavljević, Slobodan Simić and Dragan Stevanović. A survey on integral graphs. Publikacije Elektrotehničkog fakulteta. Serija Matematika, (13):42–65, 2002. URL: http://www.jstor.org/stable/43666564.
[4] Andries E. Brouwer. Small integral trees. The Electronic Journal of Combinatorics, 15(1), 2008. URL: http://eudml.org/doc/129709.
[5] Andries E. Brouwer and Wilhelmus H. Haemers. Spectra of Graphs. Springer New York, 2012. doi:10.1007/978-1-4614-1939-6.
[6] Ivan Gutman. The energy of a graph. Berichte der Mathematisch- Statistischen Sektion im Forschungszentrum Graz, 103:1–22, 1978.
[7] Ivan Gutman and Boris Furtula. Calculating vertex energies of graphs - A tutorial. MATCH Communications in Mathematical and in Computer Chemistry, 93(3):691–698, 2025. doi:10.46793/match.93-3.691G.
[8] Frank Harary and Allen J. Schwenk. Which graphs have integral spectra? In Ruth A. Bari and Frank Harary, editors, Graphs and Combinatorics, pages 45–51. Springer Berlin Heidelberg, 1974. doi:10.1007/BFb0066434.
[9] Zoran Radosavljevic and Slobodan Simic. There are just thirteen connected nonregular nonbipartite integral graphs having maximum vertex degree four (a shortened report). In Proceedings Sixth Yugoslav Seminar on Graph Theory (Dubrovnik 1985), University of Novi Sad, pages 183–187, 1986.
[10] Harishchandra S. Ramane, Sheena Y. Chowri, Thippeswamy Shiv- aprasad, and Ivan Gutman. Energy of vertices of subdivision graphs. MATCH Communications in Mathematical and in Computer Chemistry, 93(3):701–711, 2025. doi:10.46793/match.93-3.701R.
[11] Hunjanalu T. Sharathkumar, Chikkenahalli K. Shrikanth, Narahari N. Swamy, Hadonahally M. Nagesh, and Uppara Vijay Chandra Kumar. On the vertex energy of small integral trees. MATCH Communications in Mathematical and in Computer Chemistry, 95(2):467–485, 2026. doi:10.46793/match.95-2.10325.
[12] Chikkenahalli K. Shrikanth, Hunjanalu T. Sharathkumara, Narahari N. Swamy, Hadonahally M. Nagesh, and Uppara Vijay Chandra Kumar. Vertex energy of small integral graphs. Boletim da Sociedade Paranaense de Matemática, 43:1–14, 2025. doi:10.5269/bspm.77666.
Bibtex
@article{sacscuza:Nagesh2025odvecnrnbigmvdf,
title={On the Distribution of Vertex Energy of Connected Non-Regular Non-Bipartite Integral Graphs with Maximum Vertex Degree Four},
author={P. N. Simha, B. P. Shylaja, N. Narahari, H. M. Nagesh, U. V. C. Kumar},
journal={Scientific Annals of Computer Science},
volume={35},
number={2},
organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
year={2025},
pages={197-210},
publisher={Alexandru Ioan Cuza University Press, Ia\c si},
doi={10.47743/SACS.2025.2.197}
}