Browsing by Author "Togan, Müge"
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Item Alt grafların Zagreb indeksleri(Uludağ Üniversitesi, 2014-12-12) Togan, Müge; Cangül, İ. Naci; Uludağ Üniversitesi/Fen Bilimleri Enstitüsü/Matematik Anabilim Dalı.Bu çalışmada alt graflar tanıtılmış, r-alt graflar tanımlanmış ve bu alt grafların on çeşit Zagreb indeksleri hesaplanmış ve r-alt graflar için bazı eşitsizlikler verilmiştir. Bu uygulama Zagreb indekslerinin hesabında, grafların her bir köşesinin tek tek dereceleri ile uğraşmak yerine, sadece grafın kenar ve köşe sayılarının bilinmesinin yeterli olduğunu gösteren bir çalışmadır ve Zagreb indekslerinin hesabında büyük kolaylık sağlamaktadır. Bu tez dört bölümden oluşmaktadır. Birinci bölüm giriş bölümü olup bu bölümde konunun literatür özeti yapılmış ve çalışmanın ilerleyen bölümlerinde kullanılacak olan bazı temel kavramlar verilmiştir. İkinci bölümde birinci ve ikinci Zagreb indeksleri ile bunların eşindeksleri, birinci ve ikinci çarpımsal Zagreb indeksleri ile bunların eşindeksleri, total çarpımsal toplam Zagreb indeksi ile çarpımsal toplam Zagreb indeksi tanımlanarak bu indekslerin tümü için bazı sınırlar ve birbirleriyle ilişkilerini veren bazı eşitsizlikler verilmiştir. Üçüncü bölümde iyi bilinen yol graf, devir graf, yıldız graf, tam graf, iki parçalı tam graf ve tadpole grafların on çeşit Zagreb indeksleri hesaplanarak birbirleriyle ilişkilerini veren bazı sonuçlar elde edilmiştir. Dördüncü bölümde iyi bilinen bazı alt grafların ve r-alt grafların on çeşit Zagreb indeksleri hesaplanarak alt grafların çeşitli Zagreb indeksleri arasında birtakım eşitsizlikler verilmiştir. Son bölümde verilen tüm sonuçlar bu tez çalışmasında elde edilmiş orijinal sonuçlardır.Publication Inverse problem for bell index(Univ Nis, Fac Sci Math, 2020-01-01) Togan, Müge; Yurttaş, Aysun; YURTTAŞ GÜNEŞ, AYSUN; Şanlı, Utkum; Çelik, Feriha; Cangül, İsmail Naci; CANGÜL, İSMAİL NACİ; Bursa Uludağ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0001-5349-3978; 0000-0002-0700-5774; J-3505-2017; AAG-8470-2021Due to their applications in many branches of science, topological graph indices are becoming more popular every day. Especially as one can model chemical molecules by graphs to obtain valuable information about the molecules using solely mathematical calculations on the graph. The inverse problem for topological graph indices is a recent problem proposed by Gutman and is about the existence of a graph having its index value equal to a given non-negative integer. In this paper, the inverse problem for Bell index which is one of the irregularity indices is solved. Also a recently defined graph invariant called omega invariant is used to obtain several properties related to the Bell index.Item Inverse problem for sigma index(University of Kragujevac, 2018) Gutman, Ivan; Çevik, Ahmet Sinan; Togan, Müge; Yurttaş, Aysun; Naci Cangül, İsmail; Uludağ Üniversitesi/Fen - Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; GBL-2333-2022; AAG-8470-2021; J-3505-2017; 54403978300; 37090056000; 57189022403If G is a (molecular) graph and d(v), the degree of its vertex u, then its sigma index is defined as sigma(G) = Sigma(d(u) - d(v))(2), with summation going over all pairs of adjacent vertices. Some basic properties of sigma(G) are established. The inverse problem for topological indices is about the existence of a graph having its index value equal to a given non-negative integer. We study the problem for the sigma index and first show that sigma(G) is an even integer. Then we construct graph classes in which sigma(G) covers all positive even integers. We also study the inverse problem for acyclic, unicyclic, and bicyclic graphs.Item Inverse problem for Zagreb indices(Springer, 2018-10-23) Lokesha, Veerebradiah; Gutman, Ivan; Yurttaş, Aysun; Togan, Müge; Cangül, İsmail Naci; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; AAG-8470-2021; 57204472437; 54403978300; 57189022403The inverse problem for integer-valued topological indices is about the existence of a graph having its index value equal to a given integer. We solve this problem for the first and second Zagreb indices, and present analogous results also for the forgotten and hyper-Zagreb index. The first Zagreb index of connected graphs can take any even positive integer value, except 4 and 8. The same is true if one restricts to trees or to molecular graphs. The second Zagreb index of connected graphs can take any positive integer value, except 2, 3, 5, 6, 7, 10, 11, 13, 15 and 17. The same is true if one restricts to trees or to molecular graphs.Item The multiplicative Zagreb indices of graph operations(Springer, 2013) Das, Kinkar C.; Çevik, Ahmet Sinan; Yurttaş, Aysun; Togan, Müge; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; J-3505-2017; AAG-8470-2021; ABA-6206-2020; 37090056000; 54403978300; 57189022403Recently, Todeschini et al. (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: Pi(1) = Pi(1)(G) = Pi(v is an element of V(G)) d(G)(V)(2), Pi(2) = Pi(2)(G) = Pi(uv is an element of E(G)) d(G)(u)d(G)(V). These two graph invariants are called multiplicative Zagreb indices by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs. MSC: 05C05, 05C90, 05C07.Item New formulae for zagreb indices(Amer Inst Physics, 2017) Simos, T.; Tsitouras, C.; Çevik, Ahmet Sinan; Cangül, İsmail Naci; Yurttaş, Aysun; Togan, Müge; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; AAG-8470-2021; 57189022403; 37090056000; 54403978300In this paper, we study with some graph descriptors also called topological indices. These descriptors are useful in determination of some properties of chemical structures and preferred to some earlier descriptors as they are more practical. Especially the first and second Zagreb indices together with the first and second multiplicative Zagreb indices are considered and they are calculated in terms of the smallest and largest vertex degrees and vertex number for some well-known classes of graphs.Item On the first Zagreb index and multiplicative Zagreb coindices of graphs(Ovidius University, 2014-02-10) Das, Kinkar Ch; Akgüneş, Nihat; Çevik, A. Sinan; Togan, Müge; Yurttaş, Aysun; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; AAG-8470-2021; ABA-6206-2020; J-3505-2017; 54403978300; 37090056000; 57189022403For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as M-1(G) = Sigma v(i is an element of V(G))d(C)(v(i))(2), where d(G) (v(i)) is the degree of vertex v(i), in G. Recently Xu et al. introduced two graphical invariants (Pi) over bar (1) (G) = Pi v(i)v(j is an element of E(G)) (dG (v(i))+dG (v(j))) and (Pi) over bar (2)(G) = Pi(vivj is an element of E(G)) (dG (v(i))+dG (v(j))) named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = Pi(n)(i=1) d(G) (v(i)). The irregularity index t(G) of G is defined as the num=1 ber of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M-1(G) of graphs and trees in terms of number of vertices, irregularity index, maximum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and NarumiKatayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.Item Properties of n-th degree Bernstein polynomials(Amer Inst Pyhsics, 2011) Simos, T.E.; Çetin, Elif; Özbay, Hatice; Togan, Müge; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; 54402262100; 54403676000; 54403978300; 57189022403In this paper, derivatives of the product of Bernstein polynomials of the same and different degrees are obtained. Also a recurrence formula for those polynomials together with some new properties are given.Item Some formulae for the Zagreb indices of graphs(Amer Inst Physics, 2012) Çevik, Ahmet Sinan; Simos, T. E.; Psihoyios, G.; Tsitouras, C.; Anastassi, Z.; Cangül, İsmail Naci; Yurttaş, Aysun; Togan, Müge; Uludaǧ Üniversitesi/Fen Edebiyat Fakültesi/Matematik Bölümü.; 0000-0002-0700-5774; 0000-0002-0700-5774; AAG-8470-2021; J-3505-2017; ABA-6206-2020; 57189022403; 37090056000; 54403978300In this study, we first find formulae for the first and second Zagreb indices and coindices of certain classical graph types including path, cycle, star and complete graphs. Secondly we give similar formulae for the first and second Zagreb coindices.Item Some special cases of the minimal polynomial of 2cos(pi/q) over q(Amer Inst Pyhsics, 2011) Simos, T. E.; Togan, Müge; Özgür, Birsen; Cangül, İsmail Naci; Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Anabilim Dalı.; 0000-0002-0700-5774; 0000-0002-0700-5774; ABA-6206-2020; J-3505-2017; ABI-4127-2020; 54403978300; 54403501400; 57189022403The number lambda(q) - 2cos pi/q, q is an element of N, q >= 3, appears in the study of Hecke groups which are Fuchsian groups of the first kind, and in the study of regular polyhedra. Here we obtained some results on the values of the minimal polynomial of this number in modulo prime p. This results help in the calculation of the congruence subgroups of the Hecke groups which is an important problem in discrete group theory.Publication The effect of edge and vertex deletion on omega invariant(Univ Belgrade, Fac Electrical Engineering, 2020-12-01) Delen, Sadık; Togan, Müge; Yurttaş, Aysun; Ana, Uğur; Cangül, İsmail Naci; Delen, Sadık; Togan, Müge; YURTTAŞ GÜNEŞ, AYSUN; Ana, Uğur; CANGÜL, İSMAİL NACİ; Bursa Uludağ Üniversitesi/Fen-Edebiyat Fakültesi/Matematik Bölümü; 0000-0001-5349-3978; 0000-0002-0700-5774; AAG-8470-2021; J-3505-2017; EUU-3205-2022; GBL-2333-2022; CBI-5098-2022Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations.In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied.