A weighted adjacency matrix 
of a simple graph is defined for a real positive symmetric
function 
on the vertex degrees 
of a graph as
![[A_f(G)]_(ij)={f(d_i,d_j) for i,j adjacent; 0 otherwise](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FWeightedAdjacencyMatrix%2FNumberedEquation1.svg) |
(1) |
(Das et al. 2018, Zheng et al. 2022).
Special cases are summarized in the following table (c.f. Zheng et al. 2022).
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.
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Cite this as:
Weisstein, Eric W. "Weighted Adjacency Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeightedAdjacencyMatrix.html