S. Pirzada

Energy of signed digraphs

In this paper we extend the concept of energy to signed digraphs and we obtain Coulsons integral formula for energy of signed digraphs. We compute formulae for energies of signed directed cycles and we show that energy of non cycle balanced signed directed cycles increases monotonically with respect to the number of vertices. We extend the concept of non-complete extended p sum (or briefly, NEPS) to signed digraphs. We construct infinite families of noncospectral equienergetic signed digraphs. Moreover, we extend McClellands inequality to signed digraphs and also obtain a sharp upper bound for the energy of a signed digraph in terms of the number of arcs. Some open problems are also given at the end.

Spectra, Energy and Laplacian Energy of Strong Double Graphs

For a graph G with vertex set \(V (G)\,=\,\{v_{1},v_{2},\cdots \,,v_{n}\}\), the strong double graph SD(G) is a graph obtained by taking two copies of G and joining each vertex v i in one copy with the closed neighbourhood N[v i ] = N(v i ) ∪{ v i } of corresponding vertex in another copy. In this paper, we study spectra, energy and Laplacian energy of the graph SD(G). We also obtain some new families of equienergetic and L-equienergetic graphs, and an infinite family of graphs G for which LE(G) < E(G). We derive a formula for the number of spanning trees of SD(G) in terms of the number of spanning trees of G.

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

Abstract For a graph G with vertex set V(G) = {v1, v2, . . . , vn}, the extended double cover G* is a bipartite graph with bipartition (X, Y), X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}, where two vertices xi and yj are adjacent if and only if i = j or vi adjacent to vj 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 neighbors 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 Gk* 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.

SCORE SEQUENCES IN ORIENTED GRAPHS

AbstractAn oriented graph is a digraph with no symmetric pairs of directed arcs and without loops. The score of a vertexvi in an oriented graph D is

p

a_{v_i }

On equienergetic signed graphs

(or simply ai)

On scores, losing scores and total scores in hypertournaments

d_{v_i }^ -