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    ON THE DISTANCE ESTRADA INDEX OF GRAPHS
    (HACETTEPE UNIV, FAC SCI, 2009) Gungor, A. Dilek; Bozkurt, S. Burcu
    The D-eigenval ues mu(1),mu(2)...,mu(n) of a connected graph G are the eigen-values of its distance matrix D. In this paper we define and investigate n the distance Estrada index of the graph G as DEE = DEE(G) = Sigma(n)(i=1) e(mu i) and obtain bounds for DEE(G) and some relation between DEE(G) and the distance energy.
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    On the distance spectral radius and the distance energy of graphs
    (TAYLOR & FRANCIS LTD, 2011) Gungor, A. Dilek; Bozkurt, S. Burca
    The D-eigenvalues {mu(1), mu(2), ... , mu(p)} of a connected graph G are the eigenvalues of its distance matrix D. The D-energy of a graph G is the sum of the absolute values of its D-eigenvalues denoted by E(D)(G). In this article, we obtain a lower bound for the largest D-eigenvalue of G and an upper bound for E(D)(G) which improve Indulal's bounds [G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Linear Algebra Appl. 430 (2009), pp. 106-113]. In the final section of the article, we give an important remark on the distance regular graphs.
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    On the normalized Laplacian eigenvalues of graphs
    (CHARLES BABBAGE RES CTR, 2015) Das, Kinkar Ch.; Gungor, A. Dilek; Bozkurt, S. Burcu
    Let G = (V, E) be a simple connected graph with n vertices and m edges. Further let lambda(i)(L), i = 1, 2, ..., n, be the non-increasing eigenvalues of the normalized Laplacian matrix of the graph G. In this paper, we obtain the following result: For a connected graph G of order n, lambda(2)(L) = lambda(3)(L) = ... = lambda(n-1)(L) if and only if G is a complete graph K-n or G is a complete bipartite graph K-p,K- q. Moreover, we present lower and upper bounds for the normalized Laplacian spectral radius of a graph and characterize graphs for which the lower or upper bounds is attained.