Arithmetic-Geometric Matrix

The arithmetic-geometric matrix A_(AG) of a simple graph is a weighted adjacency matrix with weight


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

 [A_(AG)]_(ij)={(d_i+d_j)/(2sqrt(d_id_j))   for i,j adjacent; 0   otherwise

(Zheng et al. 2022).

Its largest eigenvalue is called the arithmetic-geometric spectral radius, half the sum of its matrix elements is the arithmetic-geometric index, and the sum of absolute values of its eigenvalues is the arithmetic-geometric energy.

See also

Adjacency Matrix, Arithmetic-Geometric Energy, Arithmetic-Geometric Index, Arithmetic-Geometric Spectral Radius, Weighted Adjacency Matrix

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Shegehall, V. S. and Kanabur, R. "Arithmetic-Geometric Indices of Path Graph." J. Math. Comput. Sci. 16, 19-24, 2015.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. "Arithmetic-Geometric Matrix." From MathWorld--A Wolfram Web Resource.

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