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Weighted Adjacency Matrix

A weighted adjacency matrix A_f of a simple graph is defined for a real positive symmetric function f(d_i,d_j) on the vertex degrees d_i of a graph as

[A_f(G)]_(ij)={f(d_i,d_j) for i,j adjacent; 0 otherwise
(1)

(Das et al. 2018, Zheng et al. 2022).

Special cases are summarized in the following table (c.f. Zheng et al. 2022).

weighted adjacency matrixf(d_i,d_j)
ABC matrixsqrt((d_i+d_j-2)/(d_id_j))
adjacency matrix1
arithmetic-geometric matrix(d_i+d_j)/(2sqrt(d_id_j))
Randić matrix1/(sqrt(d_id_j))
second Zagreb index matr ixd_id_j
Sombor matrixsqrt(d_i^2+d_j^2)

See also

Adjacency Matrix, ABC Matrix, Arithmetic-Geometric Matrix, Randić Matrix, Sombor Matrix

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References

Das, K.; Gutman, I.; Milovanović, I.; Milovanović, E.; and Furtula, B. "Degree-Based Energies of Graphs." Linear Algebra Appl. 554, 185-204, 2018.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. "Weighted Adjacency Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeightedAdjacencyMatrix.html

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