Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum.  construct infinite pairs of $D$-cospectral graphs with different diameter and different Wiener index. A graph is $k$-transmission regular if its distance matrix has constant row sum equal to $k$.  establish tight upper and lor bounds for the row sum of a $k$-transmission regular graphs in terms of the number of vertices of the graph. Finally,  determine the Wiener index and its complexity for linear $k$-trees, and obtain a closed form for the Wiener index of block-clique graphs in terms the Laplacian eigenvalues of the graph. The latter leads to a generalization of a result for trees which was proved independently by Mohar and Merris.
['Aida Abiad', 'Boris Brimkov', 'Aysel Erey', 'Lorinda Leshock', 'Xavier Martínez-Rivera', 'Suil O', 'Sung-Yell Song', 'Jason Williford']
I am writing a research paper on a idea On the Wiener index, distance cospectrality and transmission regular graphs
[{"from": "human", "value": "I am writing a research paper on a idea On the Wiener index, distance cospectrality and transmission 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:   investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum.  construct infinite pairs of $D$-cospectral graphs with different diameter and different Wiener index. A graph is $k$-transmission regular if its distance matrix has constant row sum equal to $k$.  establish tight upper and lor bounds for the row sum of a $k$-transmission regular graphs in terms of the number of vertices of the graph. Finally,  determine the Wiener index and its complexity for linear $k$-trees, and obtain a closed form for the Wiener index of block-clique graphs in terms the Laplacian eigenvalues of the graph. The latter leads to a generalization of a result for trees which was proved independently by Mohar and Merris."}]