Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here  study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.
['Miguel Angel Fiol']
I am writing a research paper on a idea Some Spectral Characterizations of Strongly Distance-Regular Graphs
[{"from": "human", "value": "I am writing a research paper on a idea Some Spectral Characterizations of Strongly Distance-Regular Graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A graph \u0393 with diameter d is strongly distance-regular if \u0393 is distance-regular and its distance-d graph \u0393d is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph \u0393d = \u0393 (the complement of \u0393), and the antipodal distance-regular graphs. Here  study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph \u0393, with distinct eigenvalues \u03bb0 > \u03bb1 > \u03bb2 > \u03bb3, is strongly distance-regular if and only if \u03bb2 = \u22121, and \u03933 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of \u0393."}]