Computation of Gutman index of Some Transformation Graphs

Yıl 2024, , 865 - 878, 01.06.2024

Merve Çakal Gökşen Bacak Turan

https://doi.org/10.21597/jist.1366169

Öz

Graph theory is widely used today in many fields, including mathematical chemistry. In this area, numerous topological indices have been studied. Topological indices are used to define the topology of a molecular graph and relate it to various relevant properties. The most commonly used topological indices are degree-based and distance-based indices. This article examines the Gutman index, which holds great significance in chemistry. The Gutman index is a well-known topological index that simultaneously uses both degree and distance. These graphs are important because they enable the study of the properties of a graph under different transformations. It has been extensively studied in the literature and finds numerous applications in chemistry, physics, and other fields. Transformation graphs are graphs that are used to model molecular structures, physical networks, and algorithms in fields such as chemistry, computer science and physics. Original graph's symmetry, connectivity and other structural features can be revealed through these transformations. This article focuses on studying the Gutman index of some transformation graphs to provide a new topological index that can be used to examine their properties, and general formulas have been obtained.

Anahtar Kelimeler

Graph theory, Transformation Graphs, Topological index, Gutman index

Bazı Transformasyon Çizgelerin Gutman İndeksinin Hesaplanması

Öz

Çizge teorisi günümüzde birçok alanda kullanılmaktadır. Bu alanlardan biri de matematiksel kimyadır. Bu alanda birçok topolojik indeks çalışılmıştır. Topolojik indeks, bir moleküler çizgenin topolojisini tanımlamak ve bunu ilgilenilen çeşitli özelliklerle ilişkilendirmek için kullanılır. Bu topolojik indekslerin en yaygın kullanılanları derece ve uzaklık tabanlı indekslerdir. Bu makalede kimyada önemli bir yere sahip olan Gutman indeksi incelenmiştir. Gutman indeksi hem derece hem de uzaklığın aynı anda kullanıldığı, literatürde kapsamlı olarak çalışılmış iyi bilinen bir topolojik indekstir ve kimya, fizik ve diğer alanlarda birçok uygulaması vardır. Transformasyon çizgeler ise kimya, bilgisayar bilimleri ve fizik gibi alanlarda moleküler yapıları, fiziksel ağları ve algoritmaları modellemek için kullanılabilen çizgelerdir. Bu çizgeler, bir çizgenin farklı transformasyonlar altındaki özelliklerinin incelenmesini sağladıklarından önemlidir. Orijinal çizgenin simetrisi, bağlantılılığı ve diğer yapısal özellikleri, bu transformasyonlar yoluyla ortaya çıkarılabilir. Bu makalede, bazı transformasyon çizgelerin Gutman indeksi bu çizgelerin özelliklerini incelemek için kullanılabilecek yeni bir topolojik indeks sağlamak amacıyla çalışılmış ve genel formüller elde edilmiştir.

Anahtar Kelimeler

Çizge teorisi, Transformasyon Çizge, Topolojik indeks, Gutman indeksi

Kaynakça

• Asir T., Rabikka V. (2022). The Wiener index of the zerodivisor graph of Zn, Discrete Applied Mathematics, 319, 461-471.
• Alizadeh, Y., Iranmanesh, A. ve Doslic, T. (2013). Additively Weighted Harary Index of Some Composite Graphs. Discrete Mathematics, 313 (1), 26-34.
• Andova, V., Dimitrov, D., Fink, J. ve Skrekovski, R. (2012). Bounds on Gutman Index. MATCH Communications in Mathematical and in Computer Chemistry, 67, 515-524.
• Aslan, E. ve Açan, B. (2019). Transformasyon Grafların Komşu İzole Saçılım Sayısı. Journal of Natural and Applied Sciences, 23(2), 625-629.
• Bacak-Turan, G. ve Kırlangıc, A. (2013). Neighbor Integrity of Transformation Graphs. International Journal of Foundations of Computer Science, 24(3), 303-317. Bacak-Turan, G. ve Oz, E. (2017). Neighbor Rupture Degree of Transformation Graphs G^(xy-). International Journal of Foundations of Computer Science, 28 (4), 335-355. Basavanagoud, B., Patil, H. P. ve Veeragoudar, J. B. (2011). On the Block-Transformation Graphs, Graph-Equations and Diameters. International Journal of Advances in Science and Technology, 2(2).
• Chartrand, G. ve Lesniak, L. (1996). Graphs and Digraphs. California: Chapman and Hall.
• Eskiizmirliler, S., Bacak-Turan, G. ve Polat, R. (2016). Neighbor Rupture Degree of Total Graphs and Their Complements. Bulletin of the International Mathematical Virtual Institute, 6(1), 55-64.
• Fajtlowicz, S. (1987). On conjectures of Graffiti-II. Congressus Numerantium, 60, 187-197.
• Furtula, B., Graovac, A. ve Vukicevic, D. (2009). Atom–bond connectivity index of trees. Discrete Applied Mathematics, 157(13), 2828-2835.
• Gursoy, A., Ülker, A. ve Kircali Gürsoy, N. (2022). Sombor index of zero-divisor graphs of commutative rings. Analele Stiintifice Ale Universitatii Ovidius Constanta-Seria Matematica, 30(2), 231-257.
• Gutman, I. (1994). Selected properties of the Schultz molecular topological index. Journal of Chemical Information and Modeling, 34(5), 1087-1089.
• Gutman, I. ve Das, K.C. (2004). The first Zagreb index 30 years after. MATCH Communications in Mathematical and in Computer Chemistry. 50, 83-92.
• Gutman, I. (2013). Degree-Based Topological Indices. Croatica Chemica Acta, 86(4), 351-361.
• Jamil, M. K. (2017). Distance-Based Topological Indices and Double Graph. Iranian Journal of Mathematical Chemistry, 8(1), 83-91.
• Kırcalı Gürsoy, N. (2021). Computing the Forgotten Topological Index for Zero Divisor Graphs of MV-Algebras. Journal of the Institute of Science and Technology, 11(4), 3072-3085.
• Klavzar, S. ve Gutman, I. (1997). Wiener number of vertex-weighted graphs and a chemical application. Discrete Applied Mathematics, 80(1), 73-81.
• Narumi, H. ve Katayama, M. (1984). Simple topological index: a newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons. Memoirs of the Faculty of Engineering, Hokkaido University, 16 (3), 209-214.
• Plavsic, D., Nikolic, S., Trinajstic, N. ve Mihalic, Z. (1993). On the Harary index for the characterization of chemical graphs. Journal of Mathematical Chemistry, 12, 235–250.
• Randic, M. (1975). On Chracterization of molecular branching. Journal of the American Chemical Society, 97, 6609-6615. Roshini, G. R., Chandrakala, S. B. ve Sooryanarayana, B. (2020). Some Degree Based Topological Indices of Transformation Graphs. Bulletin of the International Mathematical Virtual Institute, 10(2), 225-237.
• Singh P., Bhat V.K. (2021). Adjacency matrix and Wiener index of zero divisor graph Γ (Zn), Journal of Applied Mathematics and Computing, 66, 717-732.
• Todeschini, R. ve Consonni, V. (2010). New local vertex invariants and molecular descriptors based on functions of the vertex degrees. MATCH Communications in Mathematical and in Computer Chemistry, 64, 359–372.
• Vukicevic, D. ve Furtula, B. (2009). Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. Journal of mathematical chemistry, 46, 1369-1376.
• Wardecki, D, Dołowy, M. ve Bober-Majnusz, K. (2023). Evaluation of the Usefulness of Topological Indices for Predicting Selected Physicochemical Properties of Bioactive Substances with Anti-Androgenic and Hypouricemic Activity. Molecules, 28(15), 5822.
• Wiener, H. (1947). Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1), 17 - 20.
• Wu, B. (2010). Wiener Index of Line Graphs. MATCH Communications in Mathematical and in Computer Chemistry, 64, 699-706.
• Wu, B. ve Meng, J. (2001). Basic Properties of Total Transformation Graphs. Journal of Mathematical Study, 34 (2), 109-116.