Shariefuddin Pirzada

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An oriented graph is a digraph with no symmetric pairs of directed arcs and without loops. The score of a vertex vi in an oriented graph D is avi ( or simply ai) = n − 1 + d + vi − d − vi , where d + vi and d − vi are the outdegree and indegree, respectively, of vi and n is the number of vertices in D. In this paper, we give a new proof of Avery’s theorem(More)
A k-digraph is an orientation of a multi-graph that is without loops and contains at most k edges between any pair of distinct vertices. We obtain necessary and sufficient conditions for a sequence of non-negative integers in non-decreasing order to be a sequence of numbers, called marks (k-scores), attached to vertices of a k-digraph. We characterize(More)
A signed bipartite graph is a bipartite graph in which each edge is assigned a positive or a negative sign. Let G(U, V ) be a signed bipartite graph with U = {u1, u2, · · · , up} and V = {v1, v2, · · · , vq} . Then signed degree of ui is sdeg(ui) = di = d + i − d − i , where 1 ≤ i ≤ p and d+i ( d − i ) is the number of positive(negative) edges incident with(More)
For a simple connected graph G of order n, having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, the Laplacian–energy–like invariant (LEL) and the Kirchhoff index (Kf) are defined as LEL(G) = ∑n−1 i=1 √ μi and Kf(G) = n ∑n−1 i=1 1 μi , respectively. In this paper, LEL and Kf are compared, and sufficient conditions for the inequality Kf(G) < LEL(G) are(More)