Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 3, 125 - 131 HIKARI Ltd, www.m-hikari.com Some Results on the Degree of a Vertex of a Graph with Respect to Any Vertexset M. R. Rajesh Kanna 1 , B. N. Dharmendra 1 , G. Sridhara 1,2 and R. Pradeep Kumar 3 1 Department of Mathematics Maharani’s Science College for Women J. L. B. Road, Mysore - 570 005, India mr.rajeshkanna@gmail.com, bndharma@gmail.com, srsrig@gmail.com 2 Research and Development Centre Bharathiar University, Coimbatore 641 046, India srsrig@yahoo.co.in 3 National Institute of Engineering Manandavadi road, Mysore - 570 008, India pradeepr.mysore@gmail.com Abstract In this paper we have defined the degree of any vertex of a graph with respect to any vertexset. Based on this definition we have proved some results. Also we have obtained the relation between this degree and ordinary degree. Mathematics Subject Classification: Primary 05C07 Keywords: Ordinary degree, Degree of any vertex of a graph with respect to a vertexset 1. Introduction For usual notations and terminologies we refer [1], [2]. Let G(E, V ) be a simple graph. The degree of a vertex v i ∈ V is denoted by d(v i ) or d G (v i ) and is the number of vertices of V that are adjacent to v i . The new degree concepts in graphs has been defined by S S Kamath and R S Bhat [3]. In this paper we defined the degree of any vertex of a graph with respect to any vertexset.