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ve-degree, ev-degree and First Zagreb Index Entropies of Graphs

Year 2021, Volume: 6 Issue: 2, 90 - 101, 01.06.2021

Abstract

Chellali et al. introduced two degree concepts, ve-degree and ev-degree (Chellali et al, 2017). The ve-degree of a vertex v equals to number of different edges which are incident to a vertex from the closed neighborhod of v. Moreover the ev-degree of an edge e=ab equals to the number of vertices of the union of the closed neighborhoods of a and b. The most private feature of these degree concepts is, total number of ve-degrees and total number of ev-degrees equal to first Zagreb index of the graphs for triangle-free graphs. In this paper we introduce ve-degree entropy, ev-degree entropy and investigate the relations between these entropies and the first Zagreb index entropy. Finally we obtain the maximal trees with respect to ve-degree irregularity index.

References

  • [1] V. Kraus, M. Dehmer, F. Emmert-Streib, Probabilistic inequalities for evaluating structural network measures, Inform. Sci. 288 (2014), 220–245.
  • [2] M. Dehmer, F. Emmert-Streib, M. Grabner, A computational approach to construct a multivariate complete graph invariant, Inform. Sci. 260 (2014), 200–208.
  • [3] A. Mowshowitz, M. Dehmer, Entropy and the complexity of graphs revisited, Entropy 14 (3) (2012), 559–570.
  • [4] M. Dehmer, A. Mowshowitz, A case study of cracks in the scientific enterprise: reinvention of information-theoretic measures for graphs, Complexity (2014).
  • [5] Y. Chen, K. Wu, X. Chen, C. Tang, Q. Zhu, An entropy-based uncertainty measurement approach in neighborhood systems, Inform. Sci. 279 (2014), 239–250.
  • [6] C. Wang, A. Qu, Entropy, similarity measure and distance measure of vague soft sets and their relations, Inform. Sci. 244 (2013) 92–106.
  • [7] M. Dehmer, A. Mowshowitz, A history ofgraph entropy measures, Inform. Sci. 181 (2011), 57–78.
  • [8] M. Dehmer, Information processing in complex networks: graph entropy and information functionals, Appl. Math. Comput. 201 (2008), 82–94.
  • [9] M. Dehmer, A. Mowshowitz, Y. Shi, Structural differentiation of graphs using Hosoya- based indices, PLoS ONE 9 (7) (2014) e102459.
  • [10] S. Dragomir, C. Goh, Some bounds on entropy measures in information theory, Appl. Math. Lett. 10 (1997) 23–28.
  • [11] Z. Chen, M. Dehmer, Y. Shi, A note on distance-based graph entropies, Entropy 16 (10) (2014), 5416–5427.
  • [12] J. Cao, M. Shi, L. Feng, On the edge-hyper-hamiltonian laceability of balanced hypercubes, Discuss Math. Graph Theory. 36 (2016), 805–817.
  • [13] L. Feng, J. Cao, W. Liu, S. Ding, H. Liu, The spectral radius of edge chromatic critical graphs, Linear Algebra Appl. 492 (2016) 78–88.
  • [14] G. Yu, X. Liu, H. Qu, Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs, Appl. Math. Comput. 293 (2017) 287–292.
  • [15] S. Cao, M. Dehmer, Y. Shi, Extremality of degree based graph entropies, Information Sciences 278 (2014), 22-33.
  • [16] S. Cao, M. Dehmer, Degree based entropies of networks revisited, Appl. Math. Comput. 261 (2015) 141-147.
  • [17] K. C. Das, M. Dehmer, A conjecture regarding the extremal values of graph entropy based on degree powers, Entropy18 (2016) #183.
  • [18] K. C. Das, Y. Shi, Some properties on entropies of graphs, MATCH Commun. Math. Comput. Chem. 78 (2017), 259-272.
  • [19] S. Cao, M. Dehmer, Z. Kang, Network entropies based on independent sets and matchings, Appl. Math. Comput. 307 (2017) 265–270.
  • [20] P. Wan, X. Chen, J. Tu, M. Dehmer, S. Zhang, F. Emmert-Streib, On graph entropy measures based on the number of independent sets and matchings Inform. Sci. 516 (2020), 491-504.
  • [21] I. Gutman, B. Furtula, V. Katanic, Randic index and information AKCE International Journal of Graphs and Combinatorics 15 (2018), 307–312.
  • [22] M. Eliasi, On extrenal properties of general graph entropies, MATCH Commun. Math. Comput. Chem. 79 (2018), 645–657.
  • [23] M. Ghorbani, M. Dehmer, S. Zangi, On certain aspects of graph entropies of fullerenes, MATCH Commun. Math. Comput. Chem. 81 (2019), 163–174.
  • [24] R. Kazemi, Entropy of weighted graphs with the degree-based topological indices as weights, , MATCH Commun. Math. Comput. Chem. 76 (2016), 69–80.
  • [25] M. Dehmer, Z. Chen, X. Li, Y. Shi, F. Emmert-Streib, Mathematical Foundations and Applications of Graph Entropy, Wiley-Blackwell, 2016.
  • [26] J.W. Peters, Theoretical and Algorithmic Results on Domination and Connectivity, Ph.D. Dissertation, Clemson University,1986.
  • [27] J.R. Lewis, Vertex-Edge and Edge-Vertex Domination in Graphs, Ph.D. Dissertation, Clemson University, 2007.
  • [28] A. Şahin, B. Şahin, Total edge-vertex domination, RAIRO Theoretical Informatic and Applications, 54 (2020) 1.
  • [29] M. Chellali, T.W. Haynes, S.T. Hedetniemi, T.M. Lewis, On ve-degrees and ev-degrees in graphs, Discrete Math., 340 (2017) 31−38.
  • [30] B. Horoldagva, K.C. Das, T. Selenge, On ve-degree and ev-degree of the graphs, Discrete Optimization, 31 (2019),1-7.
  • [31]S. Ediz, Predicting some physicochemical properties of octane isomers: A topological approach using ev-degree and ve-degree Zagreb indices, Int. J. Syst. Sci. Appl. Math. 2 (2017), 87−92.
  • [32] B. Şahin, S. Ediz, On ev-degree and ve-degree topological indices, Iranian J. Mathematical Chemistry, 9 (4) (2018), 263-277.
  • [33] Y. Chu, A. Rauf, M. Ishtiaq, M. K. Siddiqui, M. H. Muhammad, Topological properties of polycyclic aromatic nanostars dendrimers, Polycyclic Aromatic Compounds, in press.
  • [34] J. Zhang, M. K. Siddiqui, A. Rauf, M. Ishtiaq, On ve-degree and ev-degree based topological properties of single walled titanium dioxide nanotube, J. Cluster Science, in press.
  • [35] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (4) (1972), 535-538.
  • [36] I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.
  • [37] K.C. Das, I. Gutman, B. Horoldagva, Comparing Zagreb indices and coindices of trees, MATCH Commun. Math. Comput. Chem. 68 (2012), 189–198.
  • [38] M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219-225.

Grafların ve-derece, ev-derece ve Birinci Zagreb İndeks Entropileri

Year 2021, Volume: 6 Issue: 2, 90 - 101, 01.06.2021

Abstract

Chellali et al. introduced two degree concepts, ve-degree and ev-degree (Chellali et al, 2017). The ve-degree of a vertex v equals to number of different edges which are incident to a vertex from the closed neighborhod of v. Moreover the ev-degree of an edge e=ab equals to the number of vertices of the union of the closed neighborhoods of a and b. The most private feature of these degree concepts is, total number of ve-degrees and total number of ev-degrees equal to first Zagreb index of the graphs for triangle-free graphs. In this paper we introduce ve-degree entropy, ev-degree entropy and investigate the relations between these entropies and the first Zagreb index entropy. Finally we obtain the maximal trees with respect to ve-degree irregularity index.

References

  • [1] V. Kraus, M. Dehmer, F. Emmert-Streib, Probabilistic inequalities for evaluating structural network measures, Inform. Sci. 288 (2014), 220–245.
  • [2] M. Dehmer, F. Emmert-Streib, M. Grabner, A computational approach to construct a multivariate complete graph invariant, Inform. Sci. 260 (2014), 200–208.
  • [3] A. Mowshowitz, M. Dehmer, Entropy and the complexity of graphs revisited, Entropy 14 (3) (2012), 559–570.
  • [4] M. Dehmer, A. Mowshowitz, A case study of cracks in the scientific enterprise: reinvention of information-theoretic measures for graphs, Complexity (2014).
  • [5] Y. Chen, K. Wu, X. Chen, C. Tang, Q. Zhu, An entropy-based uncertainty measurement approach in neighborhood systems, Inform. Sci. 279 (2014), 239–250.
  • [6] C. Wang, A. Qu, Entropy, similarity measure and distance measure of vague soft sets and their relations, Inform. Sci. 244 (2013) 92–106.
  • [7] M. Dehmer, A. Mowshowitz, A history ofgraph entropy measures, Inform. Sci. 181 (2011), 57–78.
  • [8] M. Dehmer, Information processing in complex networks: graph entropy and information functionals, Appl. Math. Comput. 201 (2008), 82–94.
  • [9] M. Dehmer, A. Mowshowitz, Y. Shi, Structural differentiation of graphs using Hosoya- based indices, PLoS ONE 9 (7) (2014) e102459.
  • [10] S. Dragomir, C. Goh, Some bounds on entropy measures in information theory, Appl. Math. Lett. 10 (1997) 23–28.
  • [11] Z. Chen, M. Dehmer, Y. Shi, A note on distance-based graph entropies, Entropy 16 (10) (2014), 5416–5427.
  • [12] J. Cao, M. Shi, L. Feng, On the edge-hyper-hamiltonian laceability of balanced hypercubes, Discuss Math. Graph Theory. 36 (2016), 805–817.
  • [13] L. Feng, J. Cao, W. Liu, S. Ding, H. Liu, The spectral radius of edge chromatic critical graphs, Linear Algebra Appl. 492 (2016) 78–88.
  • [14] G. Yu, X. Liu, H. Qu, Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs, Appl. Math. Comput. 293 (2017) 287–292.
  • [15] S. Cao, M. Dehmer, Y. Shi, Extremality of degree based graph entropies, Information Sciences 278 (2014), 22-33.
  • [16] S. Cao, M. Dehmer, Degree based entropies of networks revisited, Appl. Math. Comput. 261 (2015) 141-147.
  • [17] K. C. Das, M. Dehmer, A conjecture regarding the extremal values of graph entropy based on degree powers, Entropy18 (2016) #183.
  • [18] K. C. Das, Y. Shi, Some properties on entropies of graphs, MATCH Commun. Math. Comput. Chem. 78 (2017), 259-272.
  • [19] S. Cao, M. Dehmer, Z. Kang, Network entropies based on independent sets and matchings, Appl. Math. Comput. 307 (2017) 265–270.
  • [20] P. Wan, X. Chen, J. Tu, M. Dehmer, S. Zhang, F. Emmert-Streib, On graph entropy measures based on the number of independent sets and matchings Inform. Sci. 516 (2020), 491-504.
  • [21] I. Gutman, B. Furtula, V. Katanic, Randic index and information AKCE International Journal of Graphs and Combinatorics 15 (2018), 307–312.
  • [22] M. Eliasi, On extrenal properties of general graph entropies, MATCH Commun. Math. Comput. Chem. 79 (2018), 645–657.
  • [23] M. Ghorbani, M. Dehmer, S. Zangi, On certain aspects of graph entropies of fullerenes, MATCH Commun. Math. Comput. Chem. 81 (2019), 163–174.
  • [24] R. Kazemi, Entropy of weighted graphs with the degree-based topological indices as weights, , MATCH Commun. Math. Comput. Chem. 76 (2016), 69–80.
  • [25] M. Dehmer, Z. Chen, X. Li, Y. Shi, F. Emmert-Streib, Mathematical Foundations and Applications of Graph Entropy, Wiley-Blackwell, 2016.
  • [26] J.W. Peters, Theoretical and Algorithmic Results on Domination and Connectivity, Ph.D. Dissertation, Clemson University,1986.
  • [27] J.R. Lewis, Vertex-Edge and Edge-Vertex Domination in Graphs, Ph.D. Dissertation, Clemson University, 2007.
  • [28] A. Şahin, B. Şahin, Total edge-vertex domination, RAIRO Theoretical Informatic and Applications, 54 (2020) 1.
  • [29] M. Chellali, T.W. Haynes, S.T. Hedetniemi, T.M. Lewis, On ve-degrees and ev-degrees in graphs, Discrete Math., 340 (2017) 31−38.
  • [30] B. Horoldagva, K.C. Das, T. Selenge, On ve-degree and ev-degree of the graphs, Discrete Optimization, 31 (2019),1-7.
  • [31]S. Ediz, Predicting some physicochemical properties of octane isomers: A topological approach using ev-degree and ve-degree Zagreb indices, Int. J. Syst. Sci. Appl. Math. 2 (2017), 87−92.
  • [32] B. Şahin, S. Ediz, On ev-degree and ve-degree topological indices, Iranian J. Mathematical Chemistry, 9 (4) (2018), 263-277.
  • [33] Y. Chu, A. Rauf, M. Ishtiaq, M. K. Siddiqui, M. H. Muhammad, Topological properties of polycyclic aromatic nanostars dendrimers, Polycyclic Aromatic Compounds, in press.
  • [34] J. Zhang, M. K. Siddiqui, A. Rauf, M. Ishtiaq, On ve-degree and ev-degree based topological properties of single walled titanium dioxide nanotube, J. Cluster Science, in press.
  • [35] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (4) (1972), 535-538.
  • [36] I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.
  • [37] K.C. Das, I. Gutman, B. Horoldagva, Comparing Zagreb indices and coindices of trees, MATCH Commun. Math. Comput. Chem. 68 (2012), 189–198.
  • [38] M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219-225.
There are 38 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section PAPERS
Authors

Bünyamin Şahin 0000-0003-1094-5481

Abdulgani Şahin 0000-0002-9446-7431

Publication Date June 1, 2021
Submission Date March 26, 2021
Acceptance Date May 5, 2021
Published in Issue Year 2021 Volume: 6 Issue: 2

Cite

APA Şahin, B., & Şahin, A. (2021). ve-degree, ev-degree and First Zagreb Index Entropies of Graphs. Computer Science, 6(2), 90-101.

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