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Öğe New distance-based graph invariants and relations among them(ELSEVIER SCIENCE INC, 2013) Cevik, Ahmet Sinan; Maden, Ayse DilekThe eccentricity of a vertex is the maximum distance from it to another vertex, and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we introduce average edge and average vertex-edge mean eccentricities of a graph. Moreover, relations among these eccentricities for trees are provided as well as formulas for line graphs and cartesian product of graphs. (c) 2013 Elsevier Inc. All rights reserved.Öğe On the Eccentric Connectivity Index of Unicyclic Graphs(UNIV KASHAN, FAC MATHEMATICAL SCIENCES, 2018) Nacaroglu, Yasar; Maden, Ayse DilekIn this paper, we obtain the upper and lower bounds on the eccentricity connectivity index of unicyclic graphs with perfect matchings. Also we give some lower bounds on the eccentric connectivity index of unicyclic graphs with given matching numbers. (c) 2018 University of Kashan Press. All rights reservedÖğe 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.Öğe On the spectral radius of bipartite graphs which are nearly complete(SPRINGEROPEN, 2013) Das, Kinkar Chandra; Cangul, Ismail Naci; Maden, Ayse Dilek; Cevik, Ahmet SinanFor p, q, r, s, t is an element of Z(+) with rt <= p and st <= q, let G = G(p, q; r, s; t) be the bipartite graph with partite sets U = {u(1), ..., u(p)} and V = {v(1),..., v(q)} such that any two edges u(i) and v(j) are not adjacent if and only if there exists a positive integer k with 1 <= k <= t such that (k - 1) r + 1 <= i <= kr and (k - 1) s + 1 <= j <= ks. Under these circumstances, Chen et al. (Linear Algebra Appl. 432: 606-614, 2010) presented the following conjecture: Assume that p <= q, k < p, vertical bar U vertical bar = p, vertical bar V vertical bar = q and vertical bar E(G)vertical bar = pq - k. Then whether it is true that lambda(1)(G) <= lambda(1)(G(p, q; k, 1; 1)) = root pq - k + root p(2)q(2) - 6pqk + 4pk + 4qk(2) - 3k(2)/2. In this paper, we prove this conjecture for the range min(vh is an element of V){deg v(h)} <= left perpendicular p-1/2right perpendicular.