MONOPHONIC DISTANCE
Keywords:
monophonic
Abstract
A path is monophonic if it has no chords for any two vertices and in a connected graph , and the monophonic distance is the length of the longest monophonic path in . The monophonic eccentricity of each vertex in is given by . It is demonstrated that the monophonic center of a graph exists in every graph. The subgraph created by the vertices of exhibiting minimal monophonic eccentricity is the monophonic center of . Additionally, it is demonstrated that each connected graph monophonic center is located within one of its blocks.
References
1. A. P. Santhakumaran and P. Titus, A Note on “Monophonic Distance in Graphs”, Discrete Mathematics, Algorithms and Applications, Vol. 4, No. 2 (2012).
2. Azimi, A., & Ghouchan, M. F. D. (2014). Simple graphs whose 2-distance graphs are path or cycle. Le Matematiche, 69(2), 183-191.
3. Chartrand G, Escuadro H, Zhang P. Detour distance in graphs. Journal of Combinatorial Mathematics and Combinatorial Computing. 2005; 53:75-94.
4. Chartrand G, Zhang P. Distance in graphs – Taking the long view. AKCE International Journal of Graphs and Combinatorics. 2004;1(1):1-13.
5. Delorme, C. Distance biregular bipartite graphs. European Journal of Combinatorics 15, 3 (1994), 223 – 238.
6. Fiol, M. Algebraic characterizations of bipartite distance-regular graphs. In 3rd International Workshop on Optimal Networks Topologies, IWONT.
7. Fiol, M., and Garriga, E. From local adjacency polynomials to locally pseudo-distance-regular graphs. Journal of Combinatorial Theory, Series B 71, 2 (1997), 162–183.
8. Fiol, M., Garriga, E., and Yebra, J. Locally pseudo-distanceregular graphs. Journal of Combinatorial Theory, Series B 68, 2 (1996), 179–205.
9. Goddard, W., & Oellermann, O. R. (2011). Distance in graphs. In Structural Analysis of Complex Networks (pp. 49-72). Birkhäuser Boston.
10. Goddard, W., Swart, C. S., & Swart, H. C. (2005). On the graphs with maximum distance or $ k $-diameter. Mathematica Slovaca, 55(2), 131-139.
11. Godsil, C., and Shawe-Taylor, J. Distance-regularised graphs are distance-regular or distance-biregular. Journal of Combinatorial Theory, Series B 43, 1 (1987), 14 – 24.
12. M. C. Dourado, F. Protti and J.L. Szwarcfiter, Algorithmic Aspects of Monophonic Convexity, Electronic Notes in Discrete Mathematics, 30 (2008) 177-182.
13. P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph, J. Combin. Math. Combin. Comput., 84, pp. 179-188, (2013).
14. R. B. Eggleton, P. Erd¨os and D. K. Skilton, Colouring the real line, J. Combin. Theory, Ser. B, 39(1) (1985), 86–100.
15. Santhakumaran AP, Titus P. Monophonic distance in graphs. Discrete Mathematics, Algorithms and Applications. 2011; 3(2):159-169.
16. Santhakumaran AP, Titus P. The geo-number of a graph. Ars Combinatoria. 2012;106: 65-78.
17. Santhakumaran AP, Titus P. The vertex detour number of a graph. AKCE International Journal of Graphs and Combinatorics. 2007;4(1):99-112.
18. Titus P, Balakrishnan P. The forcing vertex detour monophonic number of a graph. AKCE International Journal of Graphs and Combinatorics. 2016; DOI: 10.1016/j. akcej.2016.03.002.
19. Titus P, Ganesamoorthy K. Upper edge detour monophonic number of a graph. Proyecciones Journal of Mathematics. 2014;33(2):175-187.
20. Titus, P., & Santhakumaran, A. P. (2017). Monophonic Distance in Graphs. In Graph Theory-Advanced Algorithms and Applications. Intech Open.
2. Azimi, A., & Ghouchan, M. F. D. (2014). Simple graphs whose 2-distance graphs are path or cycle. Le Matematiche, 69(2), 183-191.
3. Chartrand G, Escuadro H, Zhang P. Detour distance in graphs. Journal of Combinatorial Mathematics and Combinatorial Computing. 2005; 53:75-94.
4. Chartrand G, Zhang P. Distance in graphs – Taking the long view. AKCE International Journal of Graphs and Combinatorics. 2004;1(1):1-13.
5. Delorme, C. Distance biregular bipartite graphs. European Journal of Combinatorics 15, 3 (1994), 223 – 238.
6. Fiol, M. Algebraic characterizations of bipartite distance-regular graphs. In 3rd International Workshop on Optimal Networks Topologies, IWONT.
7. Fiol, M., and Garriga, E. From local adjacency polynomials to locally pseudo-distance-regular graphs. Journal of Combinatorial Theory, Series B 71, 2 (1997), 162–183.
8. Fiol, M., Garriga, E., and Yebra, J. Locally pseudo-distanceregular graphs. Journal of Combinatorial Theory, Series B 68, 2 (1996), 179–205.
9. Goddard, W., & Oellermann, O. R. (2011). Distance in graphs. In Structural Analysis of Complex Networks (pp. 49-72). Birkhäuser Boston.
10. Goddard, W., Swart, C. S., & Swart, H. C. (2005). On the graphs with maximum distance or $ k $-diameter. Mathematica Slovaca, 55(2), 131-139.
11. Godsil, C., and Shawe-Taylor, J. Distance-regularised graphs are distance-regular or distance-biregular. Journal of Combinatorial Theory, Series B 43, 1 (1987), 14 – 24.
12. M. C. Dourado, F. Protti and J.L. Szwarcfiter, Algorithmic Aspects of Monophonic Convexity, Electronic Notes in Discrete Mathematics, 30 (2008) 177-182.
13. P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph, J. Combin. Math. Combin. Comput., 84, pp. 179-188, (2013).
14. R. B. Eggleton, P. Erd¨os and D. K. Skilton, Colouring the real line, J. Combin. Theory, Ser. B, 39(1) (1985), 86–100.
15. Santhakumaran AP, Titus P. Monophonic distance in graphs. Discrete Mathematics, Algorithms and Applications. 2011; 3(2):159-169.
16. Santhakumaran AP, Titus P. The geo-number of a graph. Ars Combinatoria. 2012;106: 65-78.
17. Santhakumaran AP, Titus P. The vertex detour number of a graph. AKCE International Journal of Graphs and Combinatorics. 2007;4(1):99-112.
18. Titus P, Balakrishnan P. The forcing vertex detour monophonic number of a graph. AKCE International Journal of Graphs and Combinatorics. 2016; DOI: 10.1016/j. akcej.2016.03.002.
19. Titus P, Ganesamoorthy K. Upper edge detour monophonic number of a graph. Proyecciones Journal of Mathematics. 2014;33(2):175-187.
20. Titus, P., & Santhakumaran, A. P. (2017). Monophonic Distance in Graphs. In Graph Theory-Advanced Algorithms and Applications. Intech Open.
Published
2023-09-27
How to Cite
Yaqoob, M. K. (2023). MONOPHONIC DISTANCE. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 4(9), 64-74. Retrieved from https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/521
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