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Randić Matrix

The Randić matrix A_(Randic) of a simple graph is a weighted adjacency matrix with weight

f(d_i,d_j)=1/(sqrt(d_id_j)),
(1)

where d_i are the vertex degrees of the graph. In other words,

[A_(Randic)]_(ij)={1/(sqrt(d_id_j)) for i,j adjacent; 0 otherwise
(2)

(Zheng et al. 2022).

The largest eigenvalue of the Randić matrix is 0 for an empty graph and 1 otherwise, meaning its spectral radius is trivial. Half the sum of the matrix elements of the Randić matrix is the Randić index and the sum of absolute values of its eigenvalues is the Randić energy.

See also

Randić Energy, Randić Index, Randić Spectral Radius, Weighted Adjacency Matrix

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References

Bozkurt, S. B. and Guüngör, A. D. "Randić Matrix and Randić Energy." MATCH Commun. Math. Comput. Chem. 64, 239-250, 2010.Bozkurt, S. B.; Guüngör, A. D.; and Gutman, I. "Randić Spectral Radius and Randić Energy." MATCH Commun. Math. Comput. Chem. 64, 321-334, 2010.Randić, M. "On Characterization of Molecular Branching." J. Amer. Chem. Soc. 97, 6609-6615, 1975.Rodríguez, J. A. "A Spectral Approach to the Randić Index." Linear Algebra Appl. 400, 339-344, 2005.Rodríguez, J. A. and Sigarreta, J. M. "On the Randić Index and Conditional Parameters of a Graph." MATCH Commun. Math. Comput. Chem. 54, 403-416, 2005.Zheng, R.; Su, P.; and Jin. S. "Arithmetic-Geometric Matrix of Graphs and Its Applications." Appl. Math. Comput. 42, 127764, 1-11, 2023.

Cite this as:

Weisstein, Eric W. "Randić Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RandicMatrix.html

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