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    The multiplicative Zagreb indices of graph operations
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2013) Das, Kinkar C.; Yurttas, Aysun; Togan, Muge; Cevik, Ahmet Sinan; Cangul, Ismail Naci
    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.
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    The number of spanning trees of a graph
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2013) Das, Kinkar C.; Cevik, Ahmet S.; Cangul, Ismail N.
    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.
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    On the Harary index of graph operations
    (SPRINGEROPEN, 2013) Das, Kinkar C.; Xu, Kexiang; Cangul, Ismail Naci; Cevik, Ahmet Sinan; Graovac, Ante
    The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, expressions for the Harary indices of the join, corona product, Cartesian product, composition and disjunction of graphs are derived and the indices for some well-known graphs are evaluated. In derivations some terms appear which are similar to the Harary index and we name them the second and third Harary index.
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    On the Kirchhoff matrix, a new Kirchhoff index and the Kirchhoff energy
    (SPRINGEROPEN, 2013) Maden, Ayse Dilek; Cevik, Ahmet Sinan; Cangul, Ismail Naci; Das, Kinkar C.
    The main purpose of this paper is to define and investigate the Kirchhoff matrix, a new Kirchhoff index, the Kirchhoff energy and the Kirchhoff Estrada index of a graph. In addition, we establish upper and lower bounds for these new indexes and energy. In the final section, we point out a new possible application area for graphs by considering this new Kirchhoff matrix. Since graph theoretical studies (including graph parameters) consist of some fixed point techniques, they have been applied in the 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.
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    Some properties on the lexicographic product of graphs obtained by monogenic semigroups
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2013) Akgüneş, Nihat; Das, Kinkar C.; Çevik, Ahmet Sinan; Cangül, İsmail Naci
    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)).