Abstract
The distance Randić matrix of a connected graph G, denoted by \( D^{R}(G) \), was defined in Díaz and Rojo (Bull Braz Math Soc New Ser, 2021) as follows
where D(G) is the distance matrix and Tr(G) is the diagonal matrix of the transmission degrees of G. The matrix \( D^{R}(G) \) is real symmetric and the set of its eigenvalues including multiplicities is the distance Randić spectrum (or \( D^{R}\)-spectrum) of G. In the present article, we find several interesting properties of the eigenvalues of the distance Randić matrix of G. We characterize the graphs with two distance Randić eigenvalues and partially characterize the graphs with three distinct distance Randić eigenvalues. We find some new upper and lower bounds for the distance Randić energy of G and characterize the graphs attaining these bounds. One of our results improve a known upper bound of Díaz and Rojo (Bull Braz Math Soc New Ser, 2021) .
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Ganie, H.A., Rather, B.A. On Spectra of Distance Randić Matrix of Graphs. Bull Braz Math Soc, New Series 53, 1449–1467 (2022). https://doi.org/10.1007/s00574-022-00312-w
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DOI: https://doi.org/10.1007/s00574-022-00312-w