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Öğe Analysis of Zagreb indices over zero-divisor graphs of commutative rings(World Scientific Publ Co Pte Ltd, 2019) Aykac, Sumeyye; Akgunes, Nihat; Cevik, Ahmet SinanIn this paper, first Zagreb index, second Zagreb index, first multiplicative Zagreb index, second multiplicative Zagreb index, first Zagreb coindices index, second Zagreb coindices index, first multiplicative Zagreb coindices index, second multiplicative Zagreb coindices index of Gamma(Z(p2) x Z(q2)) have been established, where p and q are prime.Öğe Introducing New Exponential Zagreb Indices for Graphs(Hindawi Ltd, 2021) Akgunes, Nihat; Aydin, BusraNew graph invariants, named exponential Zagreb indices, are introduced for more than one type of Zagreb index. After that, in terms of exponential Zagreb indices, lists on equality results over special graphs are presented as well as some new bounds on unicyclic, acyclic, and general graphs are obtained. Moreover, these new graph invariants are determined for some graph operations.Öğe Line Graphs of Monogenic Semigroup Graphs(Hindawi Ltd, 2021) Akgunes, Nihat; Nacaroglu, Yasar; Pak, SedatThe concept of monogenic semigroup graphs G( SM) is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L(G( SM)) of G( SM). In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L(G( SM)).Öğe A Note on the Upper Bound of Average Distance via Irregularity Index(Springer International Publishing Ag, 2019) Akgunes, Nihat; Cangul, Ismail Naci; Cevik, Ahmet Sinan[Abstract Not Availabe]Öğe On the dot product of graphs over monogenic semigroups(Elsevier Science Inc, 2018) Akgunes, Nihat; Cagan, BusraNow define S a cartesian product of finite times with S-M(n) which is a finite semigroup having elements {0, x, x(2),..., x(n)} of order n. Gamma(S) is an undirected graph whose vertices are the nonzero elements of S. It is a new graph type which is the dot product. k be finite positive integer for 0 <={i(t)}(t=1)(k), {j(t)}(t=1)(k) <= n, any two distinct vertices of S (x(i1), x(i2),..., x(ik)) and (x(j1), x(j2),..., x(jk)) are adjacent if and only (x(i1), x(i2),..., x(ik)) . (x(j1), x(j2),..., x(jk))=0(SMn) (under the dot product) and it is assumed x(it) =0(SMn) if i(t)=0. In this study, the value of diameter, girth, maximum and minimum degrees, domination number, clique and chromatic numbers and in parallel with perfectness of Gamma(S) are elucidated. (C) 2017 Elsevier Inc. All rights reserved.Öğe On the first Zagreb index and multiplicative Zagreb coindices of graphs(Ovidius Univ Press, 2016) Das, Kinkar Ch; Akgunes, Nihat; Togan, Muge; Yurttas, Aysun; Cangul, I. Naci; Cevik, A. SinanFor 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.Öğe On the first Zagreb index and multiplicative Zagreb coindices of graphs(Ovidius Univ Press, 2016) Das, Kinkar Ch; Akgunes, Nihat; Togan, Muge; Yurttas, Aysun; Cangul, I. Naci; Cevik, A. SinanFor 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.Öğe On the Wiener index of the dot product graph over monogenic semigroups(New York Business Global Llc, 2020) Aydin, Busra; Akgunes, Nihat; Cangul, Ismail NaciAlgebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving applications. The Wiener index is one of these indices which is equal to the sum of distances between all pairs of vertices in a connected graph. The graph over the finite dot product of monogenic semigroups has recently been defined and in this paper, some results on the Wiener index of the dot product graph over monogenic semigroups are given.Öğe Some properties of zero divisor graph obtained by the ring Zp x Zq x Zr(World Scientific Publ Co Pte Ltd, 2019) Akgunes, Nihat; Nacaroglu, YasarThe concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra 116 (1988) 208-226]. In this paper, we present some properties of zero divisor graphs obtained from ring Z(p) x Z(q) x Z(r), where p, q and r are primes. Also, we give some degree-based topological indices of this special graph.Öğe Some properties on the tensor product of graphs obtained by monogenic semigroups(Elsevier Science Inc, 2014) Akgunes, Nihat; Das, Kinkar Ch.; Cevik, A. SinanIn Das et al. (2013) [8], a new graph 1'(S-M) on monogenic semigroups S-M (with zero) having elements {0, x, x(2), x(3),..., x(n)} has been recently defined. The vertices are the non-zero elements x; x(2); x(3);..., x(n) and, for 1 <= i,j <= n, any two distinct vertices x(i) and x(j) are adjacent if x(i)x(j) = 0 in S-M. As a continuing study, in Akgunes et al. (2014) [3], it has been investigated some well known indices (first Zagreb index, second Zagreb index, Randic index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over Gamma(S-M). In the light of above references, our main aim in this paper is to extend these studies over Gamma(S-M) to the tensor product. In detail, we will investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the tensor product of any two (not necessarily different) graphs Gamma(S-M)(1) and Gamma(S-M(2)). (C) 2014 Published by Elsevier Inc.
