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ABC Matrix

The ABC (atom-bond connectivity) matrix A_(ABC) of a simple graph is a weighted adjacency matrix with weight

f(d_i,d_j)=sqrt((d_i+d_j-2)/(d_id_j)),
(1)

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

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

(Zheng et al. 2022).

It was introduced by Estrada et al. (2017) to model the enthalpy of formation of alkanes (Zheng et al. 2023).

Its largest eigenvalue rho_(ABC) is called the ABC spectral radius, half the sum of its matrix elements is the ABC index, and the sum of absolute values of its eigenvalues is the ABC energy.

See also

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

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References

Estrada, E. "The ABC Matrix." J. Math. Chem. 55, 1021-1033, 2017.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. "ABC Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ABCMatrix.html

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