Energy, Laplacian energy of double graphs and new families of equienergetic graphs

For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \cdots, v_n\}$, the extended double cover $G^*$ is a bipartite graph with bipartition (X, Y), $X=\{x_1, x_2, \cdots, x_n\}$ and $Y=\{y_1, y_2, \cdots, y_n\}$, where two vertices $x_i$ and $y_j$ are adjacent if and only if $i=j$ or $v_i$ adjacent to $v_j$ in $G$. The double graph $D[G]$ of $G$ is a graph obtained by taking two copies of $G$ and joining each vertex in one copy with the neighbours of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs $G^*$ and $D[G]$, $L$-spectra of $G^{k*}$ the $k$-th iterated extended double cover of $G$. We obtain a formula for the number of spanning trees of $G^*$. We also obtain some new families of equienergetic and $L$-equienergetic graphs.Comment: 23 pages, 1 figur

Zagreb equienergetic bipartite graphs

Let G be a graph with vertices v1, v2, . . . , vn and let di be the degree of vi. The Zagreb matrix of the graph G is the square matrix of order n whose (i, j)-entry is equal to di + dj if the vertices vi and vj are adjacent, and zero otherwise. The Zagreb energy ZE(G) of G is the sum of the absolute values of the eigenvalues of the Zagreb matrix. Two graphs are said to be Zagreb equienergetic if their Zagreb energies are equal. In this paper, we show how infinitely many pairs of Zagreb equienergetic bipartite graphs can be constructed such that these bipartite graphs are connected, possess an equal number of vertices, an equal number of edges, and are not cospectral.Publisher's Versio

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

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