## Abstract

We introduce the concepts of “gravitational” and “elastic” potential energy in the context of networks and investigate their implications on interpreting certain well-known graph-theoretic parameters and optimization problems. In the case of gravitational potential energy, we treat the nodes of a graph as “particles” with masses and consider the potential energy of gravitational interactions between them. We prove that the *maximum clique* in a graph represents the minimum gravitational potential energy structure. This result yields a natural physics-based interpretation of the well-known *Motzkin-Strauss formulation* for the maximum clique problem. In the case of elastic potential energy, we assume that graph nodes are located on an elastic surface, the graph edges represent “springs” connecting the nodes on the surface, and consider the potential energy corresponding to elastic deformations of springs. We show that under certain reasonable assumptions on surface deformation, the second-smallest and the largest eigenvalues of the graph Laplacian matrix correspond to the minimum and the maximum elastic potential energies of a graph, respectively. Moreover, the associated eigenvectors correspond to the node heights in the minimum and maximum energy configurations. Note that the second-smallest eigenvalue of the graph Laplacian is widely known as *algebraic connectivity*, and it turns out that this concept also has a simple and intuitive physics-based interpretation.

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## References

- 1.
Abello, J., Pardalos, P.M., Resende, M.G.C.: On maximum clique problems in very large graphs In external memory algorithms, DIMACS series on discrete mathematics and theoretical computer science. Am. Math. Soc.

**50**, 119–130 (1999) - 2.
de Abreu, N.M.M.: Old and new results on algebraic connectivity of graphs. Linear Algebra Appl.

**423**(1), 53–73 (2007) - 3.
Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod Phys.

**74**, 47–97 (2002) - 4.
Boginski, V., Butenko, S., Pardalos, P.: Mining market data: a network approach. Comput. Oper. Res.

**33**(11), 3171–3184 (2006). doi:10.1016/j.cor.2005.01.027 - 5.
Boginski, V., Butenko, S., Pardalos, P.M.: Network models of massive datasets. Comput. Sci. Inf. Syst.

**1**(1), 75–89 (2004) - 6.
Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: handbook of combinatorial optimization, pp. 1–74. Kluwer (1999)

- 7.
Brint, A.T., Willett, P.: Algorithms for the identification of three-dimensional maximal common substructures. J. Chem. Inf. Comput. Sci.

**27**(4), 152–158 (1987) - 8.
Butenko, S., Wilhelm, W.E.: Clique-detection models in computational biochemistry and genomics. Eur. J. Oper. Res.

**173**(1), 1–17 (2006) - 9.
Chung, F.R.K.: Spectral graph theory (CBMS regional conference series in mathematics, No. 92). Am. Math. Soc. (1996).

- 10.
Fruchterman, T.M., Reingold, E.M.: Graph drawing by force-directed placement. Softw. Pract. Exp.

**21**(11), 1129–1164 (1991) - 11.
Gardiner, E.J., Artymiuk, P.J., Willett, P.: Clique-detection algorithms for matching three-dimensional molecular structures. J. Mol. Graph. Model.

**15**(4), 245–253 (1997) - 12.
Grimmett, G.R., McDiarmid, C.J.H.: On colouring random graphs. Math. Proc. Camb. Philos. Soc.

**77**, 313–324 (1976) - 13.
Grone, R., Merris, R.: The laplacian spectrum of a graph ii. SIAM J. Discret. Math.

**7**(2), 221–229 (1994) - 14.
Grone, R., Merris, R., Sunder, V.: The laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl.

**11**(2), 218–238 (1990) - 15.
Janson, S., Łuczak, T., Norros, I.: Large cliques in a power-law random graph. J. Appl. Probab.

**47**(4), 1124–1135 (2010) - 16.
Kobourov, S.G.: Spring embedders and force directed graph drawing algorithms. arXiv preprint arXiv:1201.3011 (2012)

- 17.
Luce, R.: Connectivity and generalized cliques in sociometric group structure. Psychometrika

**15**(2), 169–190 (1950) - 18.
Matula, D.W.: On the complete subgraphs of a random graph. In: proceedings of the 2nd conference on combinatorial mathematics and its applications, pp. 356–369. University of North Carolina, Chapel Hill (1970)

- 19.
Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl.

**197**, 143–176 (1994) - 20.
Mohar, B.: Some applications of laplace eigenvalues of graphs. In: graph symmetry: algebraic methods and applications, vol. 497, of NATO ASI Series C, pp. 227–275. Kluwer (1997)

- 21.
Mohar, B., Poljak, S.: Eigenvalues in combinatorial optimization. Springer (1993)

- 22.
Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math.

**17**, 533–540 (1965) - 23.
Newton, I.: Philosophiae naturalis principia mathematica (mathematical principles of natural philosophy). London (1687)

- 24.
Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE

**95**(1), 215–233 (2007) - 25.
Petroski, H.: Invention by design: how engineers get from thought to thing. Harvard University Press (1996)

- 26.
Prokopyev, O.A., Boginski, V.L., Chaovalitwongse, W., Pardalos, P.M., Sackellares, J.C., Carney, P.R.: Network-based techniques in eeg data analysis and epileptic brain modeling. In: data mining in biomedicine, pp. 559–573. Springer (2007)

- 27.
Veremyev, A., Boginski, V.: Robustness and strong attack tolerance of low-diameter networks. In: A. Sorokin, R. Murphey, M.T. Thai, P.M. Pardalos (eds.) Dynamics of information systems: mathematical foundations, Springer proceedings in mathematics and statistics, vol. 20, pp. 137–156. Springer New York (2012)

- 28.
Wasserman, S.: Social network analysis: methods and applications, vol. 8. Cambridge university press (1994)

## Acknowledgments

This research was supported in part by the Air Force Research Laboratory (AFRL) and the Defense Threat Reduction Agency (DTRA). This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. The research was performed while the first author held a National Research Council Research Associateship Award at AFRL. The authors would like to thank the referees for the constructive comments that helped to improve the paper.

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Veremyev, A., Boginski, V. & Pasiliao, E.L. Potential energy principles in networked systems and their connections to optimization problems on graphs.
*Optim Lett* **9, **585–600 (2015). https://doi.org/10.1007/s11590-014-0774-2

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### Keywords

- Networks
- Potential energy
- Graph theory
- Optimization
- Maximum clique
- Motzkin-strauss formulation
- Graph laplacian spectra
- Algebraic connectivity