Fibonacci graphs

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant omega to obtain some more information on these graphs. We give the necessary and sufficient conditions for the realizability of a set D of n successive Fibonacci numbers for every n and also list all possible realizations called Fibonacci graphs for 1 <= n <= 4

Analysis approach to finite monoids

In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal.Selçuk ÜniversitesiAkdeniz Üniversites

The multiplicative Zagreb indices of graph operations

Recently, 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.Selçuk Üniversites

The number of spanning trees of a graph

Let G be a simple connected graph of order n, m edges, maximum degree Delta(1) and minimum degree delta. Li et al. (Appl. Math. Lett. 23: 286-290, 2010) gave an upper bound on number of spanning trees of a graph in terms of n, m, Delta(1) and delta: t(G) <= delta (2m-Delta(1)-delta-1/n-3)(n-3). The equality holds if and only if G congruent to K-1,K-n-1, G congruent to K-n, G congruent to K-1 boolean OR (K-1 boolean OR Kn-2) or G congruent to K-n - e, where e is any edge of K-n. Unfortunately, this upper bound is erroneous. In particular, we show that this upper bound is not true for complete graph K-n. In this paper we obtain some upper bounds on the number of spanning trees of graph G in terms of its structural parameters such as the number of vertices (n), the number of edges (m), maximum degree (Delta(1)), second maximum degree (Delta(2)), minimum degree (delta), independence number (alpha), clique number (omega). Moreover, we give the Nordhaus-Gaddum-type result for number of spanning trees.Faculty research Fund, Sungkyunkwan UniversityKorean Government (2013R1A1A2009341)Selçuk ÜniversitesiGlaucoma Research FoundationHong Kong Baptist Universit

A new graph based on the semi-direct product of some monoids

In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of (PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics

Some fixed-point results on (generalized) Bruck-Reilly-extensions of monoids

In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly *-extensions of arbitrary monoids to be regular, coregular and strongly pi-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal.Selçuk Üniversitesi (13701071

Some properties on the lexicographic product of graphs obtained by monogenic semigroups

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph Gamma (S-M) on monogenic semigroups S-M (with zero) having elements {0, x, x(2), x(3),..., x(n)} was 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 an unpublished work, 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) were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over Gamma (S-M). In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs Gamma (S-M(1)) and Gamma (S-M(2)).Selçuk ÜniversitesiSungkyunkwan University (BK21