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Öğe A New Method for the Sum-Edge Characteristic Polynomials of Graphs(Boletim da Sociedade Paranaense de Matematica, 2022) Öz, Mert Sinan; Cangul, Ismail NaciIn this paper, the determinant of the sum-edge adjacency matrix of any given graph without loops is calculated by means of an algebraic method using spanning elementary subgraphs and also the coefficients of the corresponding sum-edge characteristic polynomial are determined by means of the elementary subgraphs. Also, we provide a formula for calculating the number of smallest odd-sized cycles in a given regular graph.Öğe Bounds for matching number of fundamental realizations according to new graph invariant omega(Jangjeon Research Institute for Mathematical Sciences and Physics, 2020) Öz, Mert Sinan; Cangul, I.N.Matching number of a graph is one of the intensively studied areas in graph theory due to numerous applications of the matching and related notions. Recently, Delen and Cangul defined a new graph invariant denoted by ? which helps to determine several graph theoretical and combinatorial properties of the realizations of a given degree sequence. In this paper, using K2 deletion process, the maximum and minimum matching numbers of all so-called fundamental realizations of a given degree sequence. © 2020 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.Öğe Computing the Hosoya and the Merrifield-Simmons Indices of Two Special Benzenoid Systems(University of Kashan, 2021) Öz, Mert Sinan; Cangul, Ismail NaciGutman et al. gave some relations for computing the Hosoya indices of two special benzenoid systems Rn and Pn. In this paper, we compute the Hosoya index and Merrifield-Simmons index of Rn and Pn by means of introducing four vectors for each benzenoid system and index. As a result, we compute the Hosoya index and the Merrifield-Simmons index of Rn and Pn by means of a product of a certain matrix of degree n and a certain vector.Öğe Computing the Merrifield-Simmons indices of benzenoid chains and double benzenoid chains(Springer Science and Business Media Deutschland GmbH, 2022) Öz, Mert Sinan; Cangul I.N.In this paper, we introduce the Merrifield-Simmons vector defined at a path of corresponding double hexagonal (benzenoid) chain. By utilizing this vector, we present reduction formulae to compute the Merrifield-Simmons index σ(H) of the corresponding double hexagonal (benzenoid) chain H. As the result, we compute σ(H) of H by means of a product of some of obtained six matrices and a vector with entries in N. Subsequently, we introduce the simple Merrifield-Simmons vector defined at an edge of given graph G. By using simple Merrifield-Simmons vector we present reduction formulae to compute the σ(G) where G represents any hexagonal (benzenoid) chain.Öğe EDGE-ZAGREB INDICES OF GRAPHS(Turkic World Mathematical Soc, 2020) Yamac, C.; Öz, Mert Sinan; Cangul, I. N.The algebraic study of graph matrices is an important area of Graph Theory giving information about the chemical and physical properties of the corresponding molecular structure. In this paper, we deal with the edge-Zagreb matrices defined by means of Zagreb indices which are the most frequently used graph indices.Öğe Matching number and characteristic polynomial of a graph(Taylor & Francis Ltd, 2020) Yurttas Gunes, Aysun; Demirci, Musa; Öz, Mert Sinan; Cangul, Ismail NaciMatching number and the spectral properties depending on the characteristic polynomial of a graph obtained by means of the adjacency polynomial has many interesting applications in different areas of science. There are some work giving the relation of these two areas. Here the relations between these two notions are considered and several general results giving this relations are obtained. A result given for only unicyclic graphs is generalized. There are some methods for determining the matching number of a graph in literature. Usually nullity, spanning trees and several graph parts are used to do this. Here, as a new method, the conditions for calculating the matching number of a graph by means of the coefficients of the characteristic polynomial of the graph are determined. Finally some results on the matching number of graphs are obtained.Öğe Matching number in relation with maximal-minimal nullity conditions and cyclomatic number by coefficient relations(Jangjeon Mathematical Society, 2019) Öz, Mert Sinan; Cangul, I.N.Let G be a simple graph. So called K2 deletion process was recently introduced by Wang. A subgraph G' of G that is obtained as a result of some K i deletion process will be called as a crucial subgroup. Let f (G) and v(G') be the matching numbers of G and G', respectively. In this study, we study the relation between i/(G), v{G') and the coefficients of the characteristic polynomials of G and G'. Several results are obtained on these notions. Moreover, conservation of maximal and minimal nullity conditions after applying Ki deletion process are studied. As a result of this, when G satisfies the maximal or minimal nullity condition, we obtain the conditions for the equality c(G) = c(G') where c(G) and c(G') denote the cyclomatic numbers of G and G', respectively. Finally, for some graphs, we state u{G) in terms of c(G), c(G'), n(G), n(G') and the coefficients of the characteristic polynomials of G and G' where n(G), n(G') are the numbers of vertices of G and G', respectively. © 2019 Jangjeon Mathematical Society. All rights reserved.Öğe Sum-edge characteristic polynomials of graphs(Taylor & Francis Ltd, 2019) Öz, Mert Sinan; Yamac, Cilem; Cangul, Ismail NaciModelling a chemical compound by a (molecular) graph helps us to obtain some required information about the chemical and physical properties of the corresponding molecular structure. Linear algebraic notions and methods are used to obtain several properties of graphs usually by the help of some graph matrices and these studies form an important sub area of Graph Theory called spectral graph theory. In this paper, we deal with the sum-edge matrices defined by means of vertex degrees. We calculate the sum-edge characteristic polynomials of several important graph classes by means of the corresponding sum-edge matrices.
