Ş. Burcu Bozkurt

Bounds on the Distance Energy and the Distance Estrada Index of Strongly Quotient Graphs

The notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada index of CSQG whose diameter does not exceed two. Additionally, we show that our results improve most of the results obtained by Gungor and Bozkurt (2009) and Zaferani (2008).

On the distance spectral radius and the distance energy of graphs

The D-eigenvalues {μ1, μ2, … , μ 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 Indulals 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.

On the complex factorization of the Lucas sequence

In this paper, firstly we present a connection between determinants of tridiagonal matrices and the Lucas sequence. Secondly, we obtain the complex factorization of Lucas sequence by considering how the Lucas sequence can be connected to Chebyshev polynomials by determinants of a sequence of matrices.

On the Number of Spanning Trees of Graphs

We establish some bounds for the number of spanning trees of connected graphs in terms of the number of vertices (n), the number of edges (m), maximum vertex degree (Δ1), minimum vertex degree (δ), first Zagreb index (M 1), and Randić index (R −1).

Upper bounds for the number of spanning trees of graphs

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.MSC:05C05, 05C50.

On the spectral radius and the energy of a digraph

The energy of a digraph is defined as , where are the (possibly complex) eigenvalues of . In this paper, we obtain an improved lower bound on the spectral radius of . Considering this result, we present an upper bound on the energy of . We also show that our results generalize and improve some known results for graphs and digraphs.