Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let Cd,kCd,k be the largest number of vertices in a Cayley digraph of degree dd and diameter kk, and let BCd,kBCd,k be the largest order of a bipartite Cayley digraph for given dd and kk. For every degree d≥2d≥2 and for every odd kk  construct Cayley digraphs of order 2k⌊d2⌋k and diameter at most kk, where k≥3k≥3, and bipartite Cayley digraphs of order 2(k−1)⌊d2⌋k−1 and diameter at most kk, where k≥5k≥5. These constructions yield the bounds Cd,k≥2k⌊d2⌋k for odd k≥3k≥3 and d≥3k2k+1, and BCd,k≥2(k−1)⌊d2⌋k−1 for odd k≥5k≥5 and d≥3k−1k−1+1. Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter k≥5k≥5. In our proofs  use new techniques based on properties of group automorphisms of direct products of abelian groups.
['Marcel Abas', 'Tomáš Vetrík']
I am writing a research paper on a idea Large Cayley digraphs and bipartite Cayley digraphs of odd diameters
[{"from": "human", "value": "I am writing a research paper on a idea Large Cayley digraphs and bipartite Cayley digraphs of odd diameters"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let Cd,kCd,k be the largest number of vertices in a Cayley digraph of degree dd and diameter kk, and let BCd,kBCd,k be the largest order of a bipartite Cayley digraph for given dd and kk. For every degree d\u22652d\u22652 and for every odd kk  construct Cayley digraphs of order 2k\u230ad2\u230bk and diameter at most kk, where k\u22653k\u22653, and bipartite Cayley digraphs of order 2(k\u22121)\u230ad2\u230bk\u22121 and diameter at most kk, where k\u22655k\u22655. These constructions yield the bounds Cd,k\u22652k\u230ad2\u230bk for odd k\u22653k\u22653 and d\u22653k2k+1, and BCd,k\u22652(k\u22121)\u230ad2\u230bk\u22121 for odd k\u22655k\u22655 and d\u22653k\u22121k\u22121+1. Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter k\u22655k\u22655. In our proofs  use new techniques based on properties of group automorphisms of direct products of abelian groups."}]