Abstract

## The Upper Edge Fixing Edge-to-edge Monophonic Number of a Graph

F. Merlin Sugirtha1*, P. Arul Paul Sudhahar2 and S. Robinson Chellathurai3

Scott Christian College, Nagercoil-629 003, INDIA. 2Assistant Professor, Department of Mathematics, Rani Anna Government Arts College (W), Tirunelveli-627 012, INDIA. 3Associate Professor, Department of Mathematics, Scott Christian College, Nagercoil-629 003, INDIA. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, INDIA.

### ABSTRACT

Let ðš be a connected graph and ð be an edge of ðš. An edge fixing edge-to-edge monophonic set ð(ð) of ðš is called a minimal edge fixing edge-to-edge monophonic set of ð of ðš, if no proper subset of ð(ð) is a minimal edge fixing edge-to-edge monophonic set ð of ðš. The maximum cardinality of a minimal edge fixing edge-to-edge monophonic set is called upper edge fixing edge-to-edge monophonic number of ð of ðš and is denoted by mðððð+(ðš). And study some of its general properties. It is shown that, for any two positive integers a and b with 2âĪaâĪb, there exists a connected graph ðš with mðððð(ðš)=a and mðððð+(ðš)=b. AMS Subject Classification: 05C12.

Keywords :edge fixing edge-to-edge monophonic number, upper edge fixing edge-to-edge monophonic number.