Some New Degree Concepts in Graphs

Kamath, S S and Bhat, R S (2006) Some New Degree Concepts in Graphs. In: International Conference on Discrete Mathematics (ICDM 2006), December 15 - 18, 2006, Ramanujan Mathematical Society and Indian Institute of Science, Bangalore - INDIA. (Submitted) PDF Some_new_degree-pub_paper.pdf - Submitted Version Restricted to Registered users only Download (235kB) | Request a copy

## Abstract

Let G = (V,E) be any graph. For any v ∈ V , the set N(v) = {u ∈ V/uv ∈ E} is the open neighbourhood of the vertex v; while the set N[v] = N(v){v} is the closed neighbourhood of v. Similarly, for any edge x = uv, N(x) = {y E/y is adjacent to x} and N[x] = N(x){x}. Then N[x] denotes the induced subgraph of G induced by the set N(v)N(u). We define three types of degrees: The VE-degree of a vertex uV , dve(u) is the number of edges in N[v] . The EV-degree of an edge x ∈ E, dev(x) is the number of vertices in N[x] and the EE-degree of an edge dee(x) is the number of edges in N[x]. In this paper, we obtain an expression for the sum of VE-degrees of all vertices, the sum of EV-degrees of all the edges and the sum of EE-degrees of all the edges, all of which are similar to the handshaking lemma. We have defined three types of domination, viz., Vertex-Edge Domination, Edge-Vertex Domination and Edge-Edge Domination. We also obtain some bounds for the three domination numbers using the new degree concepts.

Item Type: Conference or Workshop Item (Paper) VE-degree, EV-degree, EE-degree Engineering > MIT Manipal > Mathematics MIT Library 19 Oct 2011 09:35 19 Oct 2011 09:35 http://eprints.manipal.edu/id/eprint/1345 View Item