Shariefuddin Pirzada

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The set S of distinct scores (outdegrees) of the vertices of a k-partite tournament T(X1, X2, · · · , X k) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set of n non-negative integers is a score set of some k-partite(More)
The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of(More)
The imbalance of a vertex v in a digraph D is defined as a(v) = d + (v)−d − (v), where d + (v) and d − (v) respectively denote the out-degree and indegree of vertex v. The imbalance sequence of D is formed by listing vertex imbalances in nondecreasing order. We define a minimally cyclic digraph as a connected digraph which is either acyclic or has exactly(More)
The score of a vertex v in an oriented graph D is av = n−1+d + v −d − v , where d + v and d − v are the outdegree and indegree respectively of v and n is the number of vertices in D. The set of distinct scores of the vertices in an oriented graph D is called its score set. If a > 0 and d > 1 are positive integers, we show there exists an oriented graph with(More)
The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U, V) such that score set of D(U, V) is A? We conjecture that there is an affirmative answer, and verify this conjecture(More)