Reduced Second Stress Index
Resumen
The stress of a vertex is a node centrality index, which has been introduced by Shimbel (1953). The stress of a vertex in a graph is the number of geodesics (shortest paths) passing through it. A topological index of a chemical structure (graph) is a number that correlates the chemical structure with chemical reactivity or physical properties. In this paper, we introduce a new topological index for graphs called reduced second stress index using stresses of vertices. Further, we establish some inequalities, prove some results and compute reduced second stress index for some standard graphs.
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\bibitem{alf} AlFran, H. A., Somashekar, P. and Siva Kota Reddy, P.: Modified Kashvi-Tosha Stress Index for Graphs, \emph{Glob. Stoch. Anal.}, {\bf 12}(1) (2025), 10--20.
\bibitem{bdr} Bhargava, K., Dattatreya, N. N. and Rajendra, R.: On stress of a vertex in a graph, \emph{Palest. J. Math.}, {\bf 12}(3) (2023), 15–-25.
\bibitem{fge} Furtula. B., I. Gutman, and Suleyman Ediz: On Difference of Zagreb Indices, \emph{Discrete Appl. Math.}, {\bf 178} (2014), 83--88.
\bibitem{gut1} I. Gutman, N. Trinajsti\'{c}, Graph theory and molecular orbitals. Total $n$-electron energy of alternant hydrocarbons, \emph{Chem. Phys. Lett.}, 17(4) (1972), 535-538.
\bibitem{gut2} I. Gutman, B. Ru\v{s}\v{c}i\'{c}, N. Trinajsti\'{c}, C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, \emph{J. Chem. Phys.}, 62 (1975), 3399-3405.
\bibitem{hara} Harary, F.: \emph{Graph Theory}, Addison Wesley, Reading, Mass, 1972.
\bibitem{hema} Hemavathi, P. S., Lokesha, V., Manjunath, M., Siva Kota Reddy, P. and Shruti, R.: Topological Aspects of Boron Triangular Nanotube And
Boron-$\alpha$ Nanotube, \emph{Vladikavkaz Math. J}, {\bf 22}(1) (2020), 66--77.
\bibitem{kir} Kirankumar, M., Adithya, G. N., Soner Nandappa, D., Sriraj, M. A., Pavithra, M. and Siva Kota Reddy, P.: Second neighbourhood Stress Index for Graphs, \emph{Glob. Stoch. Anal.}, {\bf 12}(4) (2025), to appear.
\bibitem{raj} Mahesh, K. B., Rajendra, R. and Siva Kota Reddy, P.: Square Root Stress Sum Index for Graphs, \emph{Proyecciones}, {\bf 40}(4) (2021), 927--937.
\bibitem{indu2} Indhumathy, M., Arumugam, S., Baths, V. and Singh, T.: Graph theoretic concepts in the study of biological networks, \emph{Springer Proc. Math. Stat.}, {\bf 186} (2016), 187-200.
\bibitem{lg} Niveditha, K. Arathi Bhat, Shahistha Hanif.:Stress sum index of graphs with diameter two, \emph{Journal International Journal of Applied Mathematics}, {\bf 8}(2) (2025), 281-289.
\bibitem{pin} Pinto, R. M., Rajendra, R., Siva Kota Reddy, P. and Cangul, I. N.: A QSPR Analysis for Physical Properties of Lower Alkanes Involving Peripheral Wiener Index, \emph{Montes Taurus J. Pure Appl. Math.}, {\bf 4}(2) (2022), 81-–85.
\bibitem{pra} Prakasha, K. N., Siva kota Reddy, P. and Cangul, I. N.: Atom-Bond-Connectivity Index of Certain Graphs, \emph{TWMS J. App. Eng. Math.}, {\bf 13}(2) (2023), 400--408.
\bibitem{ma} Rajendra, R., Mahesh, K. B. and Siva Kota Reddy, P.: Mahesh Inverse Tension Index for Graphs, \emph{Adv. Math., Sci. J.}, {\bf 9}(12) (2020), 10163--10170.
\bibitem{raj1} Rajendra, R., Siva Kota Reddy, P. and Cangul, I. N.: Stress indices of graphs, \emph{Adv. Stud. Contemp. Math. (Kyungshang)}, {\bf 31}(2) (2021), 163--173.
\bibitem{raj2} Rajendra, R., Siva Kota Reddy, P. and Harshavardhana, C. N.: Tosha Index for Graphs, \emph{Proc. Jangjeon Math. Soc.}, {\bf 24}(1) (2021), 141--147.
\bibitem{rest} Rajendra, R., Siva Kota Reddy, P. and Harshavardhana, C. N.: Rest of a vertex in a graph, \emph{Adv. Math., Sci. J.}, {\bf 10}(2) (2021), 697-704.
\bibitem{rich} Rajendra, R., Siva Kota Reddy, P., Mahesh, K.B. and Harshavardhana, C. N.: Richness of a Vertex in a Graph, \emph{South East Asian J. Math. Math. Sci.}, {\bf 18}(2) (2022), 149--160.
\bibitem{raj3} Rajendra, R., Bhargava, K. Shubhalakshmi, D. and Siva Kota Reddy, P.: Peripheral Harary Index of Graphs, \emph{Palest. J. Math.}, {\bf 11}(3) (2022), 323--336.
\bibitem{raj4} Rajendra, R., Siva Kota Reddy, P. and Prabhavathi, M.: Computation of Wiener Index, Reciprocal Wiener index and Peripheral Wiener Index Using Adjacency Matrix, \emph{South East Asian J. Math. Math. Sci.}, {\bf 18}(3) (2022), 275-282.
\bibitem{raj5} Rajendra, R., Siva Kota Reddy, P., Harshavardhana, C. N., Aishwarya, S. V. and Chandrashekara, B. M.: Chelo Index for graphs, \emph{South East Asian J. Math. Math. Sci.}, {\bf19}(1) (2023), 175-188.
\bibitem{raj6} Rajendra, R., Siva Kota Reddy, P., Harshavardhana, C. N., and Alloush, Khaled A. A.: Squares Stress Sum Index for Graphs, \emph{Proc. Jangjeon Math. Soc.}, {\bf 26}(4) (2023), 483--493.
\bibitem{raj7} Rajendra, R., Siva Kota Reddy, P. and Harshavardhana, C. N.: Stress-Difference Index for Graphs, \emph{Bol. Soc. Parana. Mat. (3)}, {\bf 42} (2024), 1--10.
\bibitem{Shannon} Shannon, P., Markiel, A., Ozier, O., Baliga, N. S., Wang, J. T., Ramage, D., Amin, N., Schwikowski, B. and Ideker, T.: Cytoscape: A Software Environment for Integrated Models of Biomolecular Interaction Networks, \emph{Genome Research}, {\bf 13}(11) (2003), 2498–-2504.
\bibitem{shantha} Shanthakumari, Y., Siva Kota Reddy, P., Lokesha, V. and Hemavathi, P. S.: Topological Aspects of Boron Triangular Nanotube and Boron-Nanotube-II, \emph{South East Asian J. Math. Math. Sci.}, {\bf 16}(3) (2020), 145--156.
\bibitem{Shimbel} Shimbel, A.: Structural Parameters of Communication Networks, \emph{Bulletin of Mathematical Biophysics}, {\bf 15} (1953), 501--507.
\bibitem{siva} Siva Kota Reddy, P., Prakasha, K. N. and Cangul, I. N.: Randi\'{c} Type Hadi Index of Graphs, \emph{Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics}, {\bf 40}(4) (2020), 175--181.
\bibitem{som} Somashekar, P., Siva Kota Reddy, P., Harshavardhana, C. N. and Pavithra, M.: Cangul Stress Index for Graphs, \emph{J. Appl. Math. Inform.}, {\bf 42}(6) (2024), 1379--1388.
\bibitem{sur} Sureshkumar, S., Mangala Gowramma, H., Kirankumar, M., Pavithra, M. and Siva Kota Reddy, P.: On Maximum Stress Energy of Graphs, \emph{Glob. Stoch. Anal.}, {\bf 12}(2) (2025), 56--69.
\bibitem{HI} Xu, K., Das, K. C. and Trinajstic, N.: \emph{The Harary Index of a Graph}, Heidelberg, Springer, 2015.
\bibitem{alf} AlFran, H. A., Somashekar, P. and Siva Kota Reddy, P.: Modified Kashvi-Tosha Stress Index for Graphs, \emph{Glob. Stoch. Anal.}, {\bf 12}(1) (2025), 10--20.
\bibitem{bdr} Bhargava, K., Dattatreya, N. N. and Rajendra, R.: On stress of a vertex in a graph, \emph{Palest. J. Math.}, {\bf 12}(3) (2023), 15–-25.
\bibitem{fge} Furtula. B., I. Gutman, and Suleyman Ediz: On Difference of Zagreb Indices, \emph{Discrete Appl. Math.}, {\bf 178} (2014), 83--88.
\bibitem{gut1} I. Gutman, N. Trinajsti\'{c}, Graph theory and molecular orbitals. Total $n$-electron energy of alternant hydrocarbons, \emph{Chem. Phys. Lett.}, 17(4) (1972), 535-538.
\bibitem{gut2} I. Gutman, B. Ru\v{s}\v{c}i\'{c}, N. Trinajsti\'{c}, C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, \emph{J. Chem. Phys.}, 62 (1975), 3399-3405.
\bibitem{hara} Harary, F.: \emph{Graph Theory}, Addison Wesley, Reading, Mass, 1972.
\bibitem{hema} Hemavathi, P. S., Lokesha, V., Manjunath, M., Siva Kota Reddy, P. and Shruti, R.: Topological Aspects of Boron Triangular Nanotube And
Boron-$\alpha$ Nanotube, \emph{Vladikavkaz Math. J}, {\bf 22}(1) (2020), 66--77.
\bibitem{kir} Kirankumar, M., Adithya, G. N., Soner Nandappa, D., Sriraj, M. A., Pavithra, M. and Siva Kota Reddy, P.: Second neighbourhood Stress Index for Graphs, \emph{Glob. Stoch. Anal.}, {\bf 12}(4) (2025), to appear.
\bibitem{raj} Mahesh, K. B., Rajendra, R. and Siva Kota Reddy, P.: Square Root Stress Sum Index for Graphs, \emph{Proyecciones}, {\bf 40}(4) (2021), 927--937.
\bibitem{indu2} Indhumathy, M., Arumugam, S., Baths, V. and Singh, T.: Graph theoretic concepts in the study of biological networks, \emph{Springer Proc. Math. Stat.}, {\bf 186} (2016), 187-200.
\bibitem{lg} Niveditha, K. Arathi Bhat, Shahistha Hanif.:Stress sum index of graphs with diameter two, \emph{Journal International Journal of Applied Mathematics}, {\bf 8}(2) (2025), 281-289.
\bibitem{pin} Pinto, R. M., Rajendra, R., Siva Kota Reddy, P. and Cangul, I. N.: A QSPR Analysis for Physical Properties of Lower Alkanes Involving Peripheral Wiener Index, \emph{Montes Taurus J. Pure Appl. Math.}, {\bf 4}(2) (2022), 81-–85.
\bibitem{pra} Prakasha, K. N., Siva kota Reddy, P. and Cangul, I. N.: Atom-Bond-Connectivity Index of Certain Graphs, \emph{TWMS J. App. Eng. Math.}, {\bf 13}(2) (2023), 400--408.
\bibitem{ma} Rajendra, R., Mahesh, K. B. and Siva Kota Reddy, P.: Mahesh Inverse Tension Index for Graphs, \emph{Adv. Math., Sci. J.}, {\bf 9}(12) (2020), 10163--10170.
\bibitem{raj1} Rajendra, R., Siva Kota Reddy, P. and Cangul, I. N.: Stress indices of graphs, \emph{Adv. Stud. Contemp. Math. (Kyungshang)}, {\bf 31}(2) (2021), 163--173.
\bibitem{raj2} Rajendra, R., Siva Kota Reddy, P. and Harshavardhana, C. N.: Tosha Index for Graphs, \emph{Proc. Jangjeon Math. Soc.}, {\bf 24}(1) (2021), 141--147.
\bibitem{rest} Rajendra, R., Siva Kota Reddy, P. and Harshavardhana, C. N.: Rest of a vertex in a graph, \emph{Adv. Math., Sci. J.}, {\bf 10}(2) (2021), 697-704.
\bibitem{rich} Rajendra, R., Siva Kota Reddy, P., Mahesh, K.B. and Harshavardhana, C. N.: Richness of a Vertex in a Graph, \emph{South East Asian J. Math. Math. Sci.}, {\bf 18}(2) (2022), 149--160.
\bibitem{raj3} Rajendra, R., Bhargava, K. Shubhalakshmi, D. and Siva Kota Reddy, P.: Peripheral Harary Index of Graphs, \emph{Palest. J. Math.}, {\bf 11}(3) (2022), 323--336.
\bibitem{raj4} Rajendra, R., Siva Kota Reddy, P. and Prabhavathi, M.: Computation of Wiener Index, Reciprocal Wiener index and Peripheral Wiener Index Using Adjacency Matrix, \emph{South East Asian J. Math. Math. Sci.}, {\bf 18}(3) (2022), 275-282.
\bibitem{raj5} Rajendra, R., Siva Kota Reddy, P., Harshavardhana, C. N., Aishwarya, S. V. and Chandrashekara, B. M.: Chelo Index for graphs, \emph{South East Asian J. Math. Math. Sci.}, {\bf19}(1) (2023), 175-188.
\bibitem{raj6} Rajendra, R., Siva Kota Reddy, P., Harshavardhana, C. N., and Alloush, Khaled A. A.: Squares Stress Sum Index for Graphs, \emph{Proc. Jangjeon Math. Soc.}, {\bf 26}(4) (2023), 483--493.
\bibitem{raj7} Rajendra, R., Siva Kota Reddy, P. and Harshavardhana, C. N.: Stress-Difference Index for Graphs, \emph{Bol. Soc. Parana. Mat. (3)}, {\bf 42} (2024), 1--10.
\bibitem{Shannon} Shannon, P., Markiel, A., Ozier, O., Baliga, N. S., Wang, J. T., Ramage, D., Amin, N., Schwikowski, B. and Ideker, T.: Cytoscape: A Software Environment for Integrated Models of Biomolecular Interaction Networks, \emph{Genome Research}, {\bf 13}(11) (2003), 2498–-2504.
\bibitem{shantha} Shanthakumari, Y., Siva Kota Reddy, P., Lokesha, V. and Hemavathi, P. S.: Topological Aspects of Boron Triangular Nanotube and Boron-Nanotube-II, \emph{South East Asian J. Math. Math. Sci.}, {\bf 16}(3) (2020), 145--156.
\bibitem{Shimbel} Shimbel, A.: Structural Parameters of Communication Networks, \emph{Bulletin of Mathematical Biophysics}, {\bf 15} (1953), 501--507.
\bibitem{siva} Siva Kota Reddy, P., Prakasha, K. N. and Cangul, I. N.: Randi\'{c} Type Hadi Index of Graphs, \emph{Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics}, {\bf 40}(4) (2020), 175--181.
\bibitem{som} Somashekar, P., Siva Kota Reddy, P., Harshavardhana, C. N. and Pavithra, M.: Cangul Stress Index for Graphs, \emph{J. Appl. Math. Inform.}, {\bf 42}(6) (2024), 1379--1388.
\bibitem{sur} Sureshkumar, S., Mangala Gowramma, H., Kirankumar, M., Pavithra, M. and Siva Kota Reddy, P.: On Maximum Stress Energy of Graphs, \emph{Glob. Stoch. Anal.}, {\bf 12}(2) (2025), 56--69.
\bibitem{HI} Xu, K., Das, K. C. and Trinajstic, N.: \emph{The Harary Index of a Graph}, Heidelberg, Springer, 2015.
Publicado
2026-02-04
Sección
Research Articles
Derechos de autor 2026 Boletim da Sociedade Paranaense de Matemática
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