The Amalgamation Stability Numbers of Crown, Helm, and Book Graph

Abstract

Let G be a simple, connected, and finite graph with diam (G) > 3. Let {x, y} V (G), where d (x, y; G) 3. The 1st self-amalgamation of G with respect to x and y , denoted by G ( x, y ) is the graph obtained by identifying the vertices x and y in G. For n > 2, an nth selfamalgamation of G is recursively defined as a self-amalgamation of any n-1 selfamalgamation of G . When a self-amalgamation is no longer possible in G , it is said that G is amalgamation stable. Such a minimum natural number n for which an nth self-amalgamation of G becomes stable is referred to as the stability number of G . Results of this investigation are focused on the amalgamation stability numbers of some special graphs. Specifically, the special graphs included in this study are the Crown, Helm, and Book Graph. Keywords - Amalgamation stability numbers, crown, helm, book graph