A Study Of Edge Balanced Index Sets Of Composite Graphs Of Path And The Cycle With Chords

Graph theory is a branch of discrete mathematics,the labeling of graph is a relatively new theory in graph theory.In this paper,we study the problem of edge balance index sets extended by edge-friendly labels of graph.In 1995,the edge-friendly labeling problem is proposed and the edge balance and strong edge balance are defined by Kong and Lee in the article?On edge-balanced graph?,from then on,people began to study the problem of edge balance index sets of various graph.At present,many domestic and foreign research achievements on edge balance index sets have been made.In this paper,we study the edge balance index sets of a graph formed in series with a path which cycle graphs containing one chord and cycle.Specifically divided into:For graph(?)(m(C4,P2+{u1,u3}),P2),there are(1)when m is divided by 5,the edge balance index sets of the graph is {0,1,2,...,4m/5};(2)when m is divided by 5 and the remainder is j,the value of j is 1 or 2 or 3 or 4,the edge balance index sets of the graph is {0,1,2,...,4(m-j)/5+(j-1)}.For graph(?)(m(C4,P2+{u2,u4}),P2),the edge balance index sets is ?0,2,4,...,2m}.For graph(?)(mC4,P2),there are(1)when m is divided by 6 and the remainder is 0 or 2 or 4,the edge balance index sets of the graph is ?0,1,2,...,m};(2)when m is divided by 6 and the remainder is 1 or 3 or 5,the edge balance index sets of the graph is ?0,1,2,...,m+1}.By studying the labeling method of the edge balance index sets of the cycle and the path,and combining with the characteristics of the graphs studied,and summarize to find out the general rules.Through logical proof,the edge balance index sets of the above graph is obtained,and the correctness of the conclusion is verified by examples.