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Zagreb Indices

The first and second Zagreb indices for a graph with vertex count n and vertex degrees d_i for i=1, ..., n are defined by

Z_1(G)=sum_(i=1)^nd_i^2
(1)

and

Z_2(G)=sum_((i,j) in E(G))d_id_j,
(2)

respectively, where E(G) is the edge set of G.

The notations Z_1 (e.g., Lin et al. 2023) and M_1 (e.g., Devillers and Balaban 2000; Alikhani and Ghanbari 2024) are variously used for the first index.

According to Alikhani and Ghanbari (2024), the generalization of the first Zagreb index

F(G)=M_1^3(G)=sum_(i=1)^nd_i^3
(3)

is known as the forgotten index.

See also

First Zagreb Index, Second Zagreb Index

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References

Alikhani, S. and Ghanbari, N. "Golden Ratio in Graph Theory: A Survey." 9 Jul 2024. https://arxiv.org/abs/2407.15860.Devillers, J. and Balaban, A. T. (Eds.). "The Zabgreb Indices." In Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 28-29, 2000.Gutman, I.; Ruščić, B.; Trinajstić, N.; and Wilcox, C. F. "Graph Theory and Molecular Orbitals. XII. Acyclic Polyenes." J. Chem. Phys. 62, 3399-3409, 1975.Lin, Z.; Wang, J.; and Cai, M. "The Laplacian Spectral Ratio of Connected Graphs." 21 Feb 2023. https://arxiv.org/abs/2302.10491v1.

Cite this as:

Weisstein, Eric W. "Zagreb Indices." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZagrebIndices.html

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