17 research outputs found
On oriented graphs with minimal skew energy
Let be the skew-adjacency matrix of an oriented graph
. The skew energy of is defined as the sum of all singular
values of its skew-adjacency matrix . In this paper, we first
deduce an integral formula for the skew energy of an oriented graph. Then we
determine all oriented graphs with minimal skew energy among all connected
oriented graphs on vertices with arcs, which is an
analogy to the conjecture for the energy of undirected graphs proposed by
Caporossi {\it et al.} [G. Caporossi, D. Cvetkovi, I. Gutman, P.
Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs
with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]Comment: 15 pages. Actually, this paper was finished in June 2011. This is an
updated versio
Energy of graphs and digraphs.
Spectral graph theory (Algebraic graph theory) which emerged in 1950s and 1960s is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated to the graph. The
major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in quantum chemistry. The 1980 monograph `spectra of graphs' by Cvetkovi,c, Doob and Sach summarised nearly all research to date in the area. In 1988 it was updated by the survey `Recent results in the theory of graph spectra'. The third edition of spectra of graphs (1995) contains a summary of the
further contributions to the subject. Since then the theory has been developed to a greater extend and many research papers have been published. It is important to mention that spectral graph theory has a wide range of applications to other areas of mathematics and to other areas of sciences which include Computer Science, Physics, Chemistry, Biology, Statistics etc.Digital copy of ThesisUniversity of Kashmi
A characterization of graphs with rank 5
AbstractThe rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity n-5? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5
Laplacian energy of graphs and digraphs.
Spectral graph theory (Algebraic graph theory) which emerged in 1950s and 1960s is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues and eigenvectors of matrices associated to the graph. The major source of research in spectral graph theory has been the study of relationship between the structural and spectral properties of graphs. Another source has research in quantum chemistry. Just as astronomers study stellar spectra to determine the make-up of distant stars, one of the main goals in spectral graph theory is to deduce the principal properties and structure of a graph from its graph spectrum (or from a short list of easily computable invariants). The spectral approach for general graphs is a step in this direction.Digital copy of Thesis.University of Kashmir