Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A finite simple graph G is called a sum graph (respectively, integral sum graph) if there is a bijectionf from the vertices of G to a set of positive integers S (respectively, a set of integers S) such that uv is an edge of G if and only if f(u)+f(v)@?S. For graphs with n vertices,  show that there exist sum graphs with m edges if and only if m@?@?14(n-1)^2@? and that there exists integral sum graphs with m edges if and only if m@?@?38(n-1)^2@?+@?12(n-1)@?, except for m=@?38(n-1)^2@?+@?12(n-1)@?-1 when n is of the form 4k+1.  also characterize sets of positive integers (respectively, integers) which are in bijection with sum graphs (respectively, integral sum graphs) of maximum size for a given order.
['Apurv Tiwari', 'Amitabha Tripathi']
I am writing a research paper on a idea On the range of size of sum graphs & integral sum graphs of a given order
[{"from": "human", "value": "I am writing a research paper on a idea On the range of size of sum graphs & integral sum graphs of a given order"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A finite simple graph G is called a sum graph (respectively, integral sum graph) if there is a bijectionf from the vertices of G to a set of positive integers S (respectively, a set of integers S) such that uv is an edge of G if and only if f(u)+f(v)@?S. For graphs with n vertices,  show that there exist sum graphs with m edges if and only if m@?@?14(n-1)^2@? and that there exists integral sum graphs with m edges if and only if m@?@?38(n-1)^2@?+@?12(n-1)@?, except for m=@?38(n-1)^2@?+@?12(n-1)@?-1 when n is of the form 4k+1.  also characterize sets of positive integers (respectively, integers) which are in bijection with sum graphs (respectively, integral sum graphs) of maximum size for a given order."}]