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    Derived Graphs of Some Graphs
    (2012) Jog, R. Jog; Satısh, P. Hande; Gutman, Ivan; Bozkurt, Ş. Burcu
    The derived graph of a simple graph G, denoted by G†, is the graph having the same vertex set as G, in which two vertices are adjacent if and only if their distance in G is two. Continuing the studies communicated in Kragujevac J. Math. 34 (2010), 139-146, we examined derived graphs of some graphs and determine their spectra.
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    Note on the Distance Energy of Graphs
    (UNIV KRAGUJEVAC, FAC SCIENCE, 2010) Bozkurt, Ş. Burcu; Güngör, A. Dilek; Zhou, Bo
    The distance energy of a graph G is defined as the sum of the absolute values of the eigenvalues of the distance matrix of G. In this note, we obtain an upper bound for the:distance energy of any connected graph G. Specially, we present upper bounds for the distance energy of connected graphs of diameter 2 with given numbers of vertices and edges, and unicyclie graphs with odd girth. Additionally, we give also a lower bound for the distance energy of unicyclic graphs with odd girth.
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    Randic Matrix and Randic Energy
    (Univ Kragujevac, Fac Science, 2010) Bozkurt, Ş. Burcu; Güngör, A. Dilek; Gutman, Ivan; Çevik, A. Sinan
    If G is a graph on n vertices, and d(i) is the degree of its i-th vertex, then the Randic matrix of G is the square matrix of order n whose (i, j)-entry is equal to 1/root d(i) d(j) di if the i-th and j-th vertex of G are adjacent, and zero otherwise. This matrix in a natural way occurs within Laplacian spectral theory, and provides the non-trivial part of the so-called normalized Laplacian matrix. In spite of its obvious relation to the famous Randic index, the Randic matrix seems to have not been much studied in mathematical chemistry. In this paper we define the Randic energy as the sum of the absolute values of the eigenvalues of the Randic matrix, and establish mine of its properties, in particular lower and upper bounds for it.
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    Randic Spectral Radius and Randic Energy
    (Univ Kragujevac, Fac Science, 2010) Bozkurt, Ş. Burcu; Güngör, A. Dilek; Gutman, Ivan
    Let G be a simple connected graph with n vertices and let d(i) be the degree of its i-th vertex. The Randic matrix of G is the square matrix of order n whose (i, j)-entry is equal to 1/root d(i)d(j) di if the i-th and j-th vertex of G are adjacent, and zero otherwise. The Randic eigenvalues are the eigenvalues of the Rancho matrix. The greatest Randic eigenvalue is the Randic spectral radius of C. The Randic energy is the sum of the absolute values of the Randic eigenvalues. Lower bounds for Randic spectral radius and an upper bound for Randic energy are obtained. Graphs for which these bounds are best possible are characterized.
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    Upper Bounds for the Number of Spanning Trees of Graphs
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2012) Bozkurt, Ş. Burcu
    In this paper, we present some upper bounds for the number of spanning trees of graphs in terms of the number of vertices, the number of edges and the vertex degrees.