The many facets of the Estrada indices of graphs and networks

Abadias, L., Estrada-Rodriguez, G., Estrada, E.: Fractional-order susceptible-infected model: definition and applications to the study of covid-19 main protease. Fract. Calc. Appl. Anal. 23(3), 635–655 (2020)

Acharya, B.D.: Spectral criterion for cycle balance in networks. J. Graph Theory 4(1), 1–11 (1980)

Ala-Nissila, T., Ferrando, R., Ying, S.: Collective and single particle diffusion on surfaces. Adv. Phys. 51(3), 949–1078 (2002)

Aleksić, T., Gutman, I., Petrović, M.: Estrada index of iterated line graphs. Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathématiques), pp. 33–41 (2007)

Alhomaidhi, A., Al-Thukair, F., Estrada, E.: Gaussianization of the spectra of graphs and networks. theory and applications. J. Math. Anal. Appl. 470(2), 876–897 (2019)

Alhomaidhi, A., Al-Thukair, F., Estrada, E.: Double gaussianization of graph spectra. Appl. Math. Model. 93, 134–147 (2021)

Allen, L.J., Brauer, F., Van den Driessche, P., Wu, J.: Mathematical Epidemiology, vol. 1945. Springer, New York (2008)

Almeida, R., da Cruz, A.M.B., Martins, N., Monteiro, M.T.T.: An epidemiological MSEIR model described by the caputo fractional derivative. Int. J. Dyn. Control 7(2), 776–784 (2019)

Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)

Andrade, E., Lenes, E., Mallea-Zepeda, E., Robbiano, M., Rodríguez, J.: Extremal graphs for Estrada indices. Linear Algebra Appl. 588, 54–73 (2020)

Angstmann, C., Henry, B., McGann, A.: A fractional order recovery sir model from a stochastic process. Bull. Math. Biol. 78(3), 468–499 (2016)

Arenas, A.J., Gonzalez-Parra, G., Chen-Charpentier, B.M.: Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Math. Comput. Simul. 121, 48–63 (2016)

Arrigo, F., Durastante, F.: Mittag–Leffler functions and their applications in network science. arXiv preprint arXiv:2103.12559 (2021)

Ashrafi, A., Fath-Tabar, G.H.: Bounds on the Estrada index of ISR (4, 6)-fullerenes. Appl. Math. Lett. 24(3), 337–339 (2011)

Asratian, A.S., Denley, T.M., Häggkvist, R.: Bipartite Graphs and their Applications, vol. 131. Cambridge University Press, Cambridge (1998)

Ayyaswamy, S.K., Balachandran, S., Venkatakrishnan, Y.B., Gutman, I.: Signless Laplacian Estrada index. MATCH Commun. Math. Comput. Chem. 66(3), 785–794 (2011)

Baird, D., Ulanowicz, R.E.: The seasonal dynamics of the Chesapeake bay ecosystem. Ecol. Monogr. 59(4), 329–364 (1989)

Ballini, F., Deniskin, N.: Vertex distinction with subgraph centrality: a proof of Estrada’s conjecture and some generalizations. Linear Algebra Appl. 612, 334–344 (2021)

Bamdad, H.: New lower bounds for Estrada index. Bull. Malays. Math. Sci. Soc. 39(2), 683–688 (2016)

Bamdad, H., Ashraf, F., Gutman, I.: Lower bounds for Estrada index and Laplacian Estrada index. Appl. Math. Lett. 23(7), 739–742 (2010)

Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

Barlow, M.T.: Random Walks and Heat Kernels on Graphs, vol. 438. Cambridge University Press, Cambridge (2017)

Beaudin, L., Ellis-Monaghan, J., Pangborn, G., Shrock, R.: A little statistical mechanics for the graph theorist. Discrete Math. 310(13–14), 2037–2053 (2010)

Beineke, L.W., Wilson, R.J., Cameron, P.J., et al.: Topics in Algebraic Graph Theory, vol. 102. Cambridge University Press, Cambridge (2004)

Belyayev, Y.N.: On the calculation of functions of matrices. Math. Notes 94(1), 177–184 (2013)

Benzi, M.: A note on walk entropies in graphs. Linear Algebra Appl. 445, 395–399 (2014)

Benzi, M., Boito, P.: Quadrature rule-based bounds for functions of adjacency matrices. Linear Algebra Appl. 433(3), 637–652 (2010)

Benzi, M., Boito, P.: Matrix functions in network analysis. GAMM-Mitteilungen 43(3), e202000012 (2020)

Berge, C.: The Theory of Graphs (trans. alison doig). Methuen & Co., London (1962)

Biggs, N., Biggs, N.L., Norman, B.: Algebraic Graph Theory, vol. 67. Cambridge University Press, Cambridge (1993)

Binder, K., Landau, D.: Phase diagrams and critical behavior in Ising square lattices with nearest-and next-nearest-neighbor interactions. Phys. Rev. B 21(5), 1941 (1980)

Bouillard, A., Boyer, M., Le Corronc, E.: Deterministic Network Calculus: From Theory to Practical Implementation. Wiley (2018)

Bozkurt, ŞB., Adiga, C., Bozkurt, D.: On the energy and Estrada index of strongly quotient graphs. Indian J. Pure Appl. Math. 43(1), 25–36 (2012)

Brauer, F.: Mathematical epidemiology: past, present, and future. Infect. Dis. Model. 2(2), 113–127 (2017)

Braun, S., Ronzheimer, J.P., Schreiber, M., Hodgman, S.S., Rom, T., Bloch, I., Schneider, U.: Negative absolute temperature for motional degrees of freedom. Science 339(6115), 52–55 (2013)

Canning, A., Wang, L., Williamson, A., Zunger, A.: Parallel empirical pseudopotential electronic structure calculations for million atom systems. J. Comput. Phys. 160(1), 29–41 (2000)

Caputo, M.: Linear models of dissipation whose q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)

Carmona, J.R., Rodríguez, J.: An increasing sequence of lower bounds for the Estrada index of graphs and matrices. Linear Algebra Appl. 580, 200–211 (2019)

Chen, L., Painter, K., Surulescu, C., Zhigun, A.: Mathematical models for cell migration: a non-local perspective. Philos. Trans. R. Soc. B 375(1807), 20190379 (2020)

Chen, X., Hou, Y.: Some results on Laplacian Estrada index of graphs. MATCH Commun. Math. Comput. Chem. 73, 149–162 (2015)

Chen, X., Qian, J.: On resolvent Estrada index. MATCH Commun. Math. Comput. Chem 73, 163–174 (2015)

Chen, Y., Du, J., Xiong, M., Guo, H., Jinnai, H., Kaneko, T.: Perforated block copolymer vesicles with a highly folded membrane. Macromolecules 40(13), 4389–4392 (2007)

Chen, Z., Fan, Y.-Z., Du, W.-X.: Estrada index of random graphs. MATCH Commun. Math. Comput. Chem. 68, 825–834 (2012)

Chu, R., Pei, W., Takei, J., Bai, Y.: Relationship between the native-state hydrogen exchange and folding pathways of a four-helix bundle protein. Biochemistry 41(25), 7998–8003 (2002)

Chung, F.: The heat kernel as the pagerank of a graph. Proc. Natl. Acad. Sci. 104(50), 19735–19740 (2007)

Chung, F.R., Graham, F.C.: Spectral Graph Theory, vol. 92. American Mathematical Society, Providence (1997)

Clemente, G.P., Cornaro, A.: Novel bounds for the normalized Laplacian Estrada index and normalized Laplacian energy. MATCH Commun. Math. Comput. Chem. 77(3), 673–690 (2017)

Corli, A., Malaguti, L.: Wavefronts in traffic flows and crowds dynamics. In: Anomalies in Partial Differential Equations, pp. 167–189. Springer (2021)

Cox, L.A., Jr., Popken, D.A., Sun, R.X., et al.: Causal Analytics for Applied Risk Analysis. Springer, New York (2018)

Cvetkovic, D., Cvetković, D.M., Rowlinson, P., Simic, S.: Eigenspaces of Graphs, vol. 66. Cambridge University Press, Cambridge (1997)

Cvetkovic, D.M.: Applications of graph spectra: an introduction to the literature. Appl. Graph Spectra 13(21), 7–31 (2009)

Cvetkovic, D.M., Dragos, M. et al. Spectra of Graphs. Theory and Application (1980)

Das, K.C., Lee, S.-G.: On the Estrada index conjecture. Linear Algebra Appl. 431(8), 1351–1359 (2009)

de la Peña, J.A., Gutman, I., Rada, J.: Estimating the Estrada index. Linear Algebra Appl. 427(1), 70–76 (2007)

Deng, H.: A note on the Estrada index of trees. Match 62(3), 607 (2009)

Deng, H.: A proof of a conjecture on the Estrada index. Match 62(3), 599 (2009)

Deng, H., Radenkovic, S., Gutman, I.: The Estrada index. Applications of Graph Spectra, Math. Inst., Belgrade, pp. 123–140 (2009)

Deng, H., Zhang, J.: A note on the Laplacian Estrada index of trees. MATCH Commun. Math. Comput. Chem. 63(3) (2010)

Dias, J.R.: Molecular Orbital Calculations Using Chemical Graph Theory, vol. 19. Springer, New York (1993)

Du, Z.: More on Laplacian Estrada indices of trees. Filomat 26(1), 197–206 (2012)

Du, Z., Liu, Z.: On the Estrada and Laplacian Estrada indices of graphs. Linear Algebra Appl. 435(8), 2065–2076 (2011)

Du, Z., Zhou, B.: The Estrada index of trees. Linear Algebra Appl. 435(10), 2462–2467 (2011)

Du, Z., Zhou, B.: On the Estrada index of graphs with given number of cut edges. Electron. J. Linear Algebra 22 (2011)

Du, Z., Zhou, B.: The Estrada index of unicyclic graphs. Linear Algebra Appl. 436(9), 3149–3159 (2012)

Dung, V.B., Van Thien, D.: The equation of backward diffusion and negative diffusivity. J. Phys. Conf. Ser. 537, 012011 (2014)

Edward, J.T.: Molecular volumes and the Stokes–Einstein equation. J. Chem. Educ. 47(4), 261 (1970)

Enns, A., Becker, T.W., Schmeling, H.: The dynamics of subduction and trench migration for viscosity stratification. Geophys. J. Int. 160(2), 761–775 (2005)

Erdos, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci 5(1), 17–60 (1960)

Essam, J.W.: Graph theory and statistical physics. Discrete Math. 1(1), 83–112 (1971)

Estrada, E.: Characterization of 3d molecular structure. Chem. Phys. Lett. 319(5–6), 713–718 (2000)

Estrada, E.: Characterization of the folding degree of proteins. Bioinformatics 18(5), 697–704 (2002)

Estrada, E.: Generalized graph matrix, graph geometry, quantum chemistry, and optimal description of physicochemical properties. J. Phys. Chem. A 107(38), 7482–7489 (2003)

Estrada, E. Characterization of the amino acid contribution to the folding degree of proteins. Proteins Struct. Funct. Bioinform. 54(4), 727–737 (2004)

Estrada, E.: Point scattering: a new geometric invariant with applications from (nano) clusters to biomolecules. J. Comput. Chem. 28(4), 767–777 (2007)

Estrada, E.: Generalized walks-based centrality measures for complex biological networks. J. Theor. Biol. 263(4), 556–565 (2010)

Estrada, E.: Universality in protein residue networks. Biophys. J. 98(5), 890–900 (2010)

Estrada, E.: Path Laplacian matrices: introduction and application to the analysis of consensus in networks. Linear Algebra Appl. 436(9), 3373–3391 (2012)

Estrada, E.: The Structure of Complex Networks: Theory and Applications. Oxford University Press, Oxford (2012)

Estrada, E.: Rethinking structural balance in signed social networks. Discrete Appl. Math. 268, 70–90 (2019)

Estrada, E., Alhomaidhi, A.A., Al-Thukair, F.: Exploring the “middle earth” of network spectra via a gaussian matrix function. Chaos 27(2), 023109 (2017)

Estrada, E., Benzi, M.: Walk-based measure of balance in signed networks: detecting lack of balance in social networks. Phys. Rev. E 90(4), 042802 (2014)

Estrada, E., Gómez-Gardeñes, J.: Network bipartivity and the transportation efficiency of European passenger airlines. Phys. D Nonlinear Phenom. 323, 57–63 (2016)

Estrada, E., Hatano, N.: Statistical-mechanical approach to subgraph centrality in complex networks. Chem. Phys. Lett. 439(1–3), 247–251 (2007)

Estrada, E., Hatano, N., Benzi, M.: The physics of communicability in complex networks. Phys. Rep. 514(3), 89–119 (2012)

Estrada, E., Higham, D.J.: Network properties revealed through matrix functions. SIAM Rev. 52(4), 696–714 (2010)

Estrada, E., José, A., Hatano, N.: Walk entropies in graphs. Linear Algebra Appl. 443, 235–244 (2014)

Estrada, E., Rodríguez-Velázquez, J.A.: Spectral measures of bipartivity in complex networks. Phys. Rev. E 72(4), 046105 (2005)

Estrada, E., Rodriguez-Velazquez, J.A.: Subgraph centrality in complex networks. Phys. Rev. E 71(5), 056103 (2005)

Estrada, E., Silver, G.: Accounting for the role of long walks on networks via a new matrix function. J. Math. Anal. Appl. 449(2), 1581–1600 (2017)

Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: The Structure and Dynamics of Networks, pp. 195–206. Princeton University Press (2011)

Fath-Tabar, G., Ashrafi, A.: New upper bounds for Estrada index of bipartite graphs. Linear Algebra Appl. 435(10), 2607–2611 (2011)

Fath-Tabar, G., Ashrafi, A., Gutman, I.: Note on Estrada and l-Estrada indices of graphs. Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathématiques), pp. 1–16 (2009)

Festinger, L.: The analysis of sociograms using matrix algebra. Hum. Relat. 2(2), 153–158 (1949)

Fixman, M.: Radius of gyration of polymer chains. J. Chem. Phys. 36(2), 306–310 (1962)

Fowler, P.: Hückel spectra of möbius $$\pi $$ systems. Phys. Chem. Chem. Phys. 4(13), 2878–2883 (2002)

Fowler, P.W.: Möbius systems and the Estrada index. MATCH Commun. Math. Comput. Chem. 66, 751–764 (2011)

Frommer, A., Simoncini, V.: Matrix functions. In: Model Order Reduction: Theory, Research Aspects and Applications, pp. 275–303. Springer, New York (2008)

Fukui, K.: The role of frontier orbitals in chemical reactions (nobel lecture). Angew. Chem. Int. Ed. Engl. 21(11), 801–809 (1982)

Fukui, K., Yonezawa, T., Shingu, H.: A molecular orbital theory of reactivity in aromatic hydrocarbons. J. Chem. Phys. 20(4), 722–725 (1952)

Gao, N., Hu, D., Liu, X., Zhang, S.: Laplacian spectral moment and Laplacian Estrada index of random graphs. J. Math. Anal. Appl. 461(2), 1299–1307 (2018)

Gao, N., Qiao, L., Ning, B., Zhang, S.: Coulson-type integral formulas for the Estrada index of graphs and the skew Estrada index of oriented graphs. MATCH Commun. Math. Comput. Chem. 73, 133–148 (2015)

Garrappa, R., Popolizio, M.: Computing the matrix Mittag–Leffler function with applications to fractional calculus. J. Sci. Comput. 77(1), 129–153 (2018)

Ghorbani, M.: On the energy and Estrada index of Cayley graphs. Discrete Math. Algorithms Appl. 7(01), 1550005 (2015)

Ginosar, Y., Gutman, I., Mansour, T., Schork, M.: Estrada index and Chebyshev polynomials. Chem. Phys. Lett. 454(4–6), 145–147 (2008)

Godsil, C., Royle, G.F.: Algebraic Graph Theory, vol. 207. Springer Science & Business Media (2001)

Godsil, C.D., McKay, B.D.: Feasibility conditions for the existence of walk-regular graphs. Linear Algebra Appl. 30, 51–61 (1980)

Gordon, P.A.: Characterizing isoparaffin transport properties with Stokes–Einstein relationships. Ind. Eng. Chem. Res. 42(26), 7025–7036 (2003)

Güngör, A., Bozkurt, S.: On the distance Estrada index of graphs. Hacettepe J. Math. Stat. 38(3), 277–283 (2009)

Güngör, A.D., Cevik, A.S.: On the Harary energy and Harary Estrada index of a graph. MATCH Commun. Math. Comput. Chem. 64(1), 281–296 (2010)

Gutman, I.: Lower bounds for Estrada index. Publ. Inst. Math. 83(97), 1–7 (2008)

Gutman, I., Borovicanin, B.: Nullity of graphs: an updated survey. Zbornik Radova 14(22), 137–154 (2011)

Gutman, I., Deng, H., Radenkovic, S.: The Estrada index: an updated survey. Selected Topics on Applications of Graph Spectra, Math. Inst., Beograd, pp. 155–174 (2011)

Gutman, I., Estrada, E., Rodríguez-Velázquez, J.A.: On a graph-spectrum-based structure descriptor. Croat. Chem. Acta 80(2), 151–154 (2007)

Gutman, I., Furtula, B., Chen, X., Qian, J.: Resolvent Estrada index-computational and mathematical studies. MATCH Commun. Math. Comput. Chem. 74, 431–440 (2015)

Gutman, I., Furtula, B., Marković, V., Glišić, B.: Alkanes with greatest Estrada index. Zeitschrift für Naturforschung A 62(9), 495–498 (2007)

Gutman, I., Graovac, A.: Estrada index of cycles and paths. Chem. Phys. Lett. 436(1–3), 294–296 (2007)

Gutman, I., Medina, L., Pizarro, P., Robbiano, M.: Graphs with maximum laplacian and signless laplacian Estrada index. Discrete Math. 339(11), 2664–2671 (2016)

Gutman, I., Radenković, S.: Estrada index of benzenoid hydrocarbons. Zeitschrift für Naturforschung A 62(5–6), 254–258 (2007)

Gutman, I., Radenkovic, S.: A lower bound for the Estrada index of bipartite molecular graphs. Kragujevac J. Sci. 29, 67–72 (2007)

Gutman, I., Radenkovic, S.: A lower bound for the Estrada index of bipartite molecular graphs. Kragujevac J. Sci. 29, 67–72 (2007)

Gutman, I., Radenković, S., Furtula, B., Mansour, T., Schork, M.: Relating Estrada index with spectral radius. J. Serb. Chem. Soc. 72(12), 1321–1327 (2007)

Hakimi-Nezhaad, M., Ghorbani, M.: On the Estrada index of Seidel matrix. Math. Interdiscip. Res. 5(1), 43–54 (2020)

Hakimi-Nezhaad, M., Hua, H., Ashrafi, A.R., Qian, S.: The normalized Laplacian Estrada index of graphs. J. Appl. Math. Inform. 32(1–2), 227–245 (2014)

Hall, G.: On the eigenvalues of molecular graphs. Mol. Phys. 33(2), 551–557 (1977)

Hambly, B.M., Kumagai, T.: Diffusion processes on fractal fields: heat kernel estimates and large deviations. Probab. Theory Relat. Fields 127(3), 305–352 (2003)

He, L., Niemeyer, B.: A novel correlation for protein diffusion coefficients based on molecular weight and radius of gyration. Biotechnol. Prog. 19(2), 544–548 (2003)

Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)

Hofman, M.A.: Size and shape of the cerebral cortex in mammals (part 1 and 2). Brain Behav. Evol. 27(1), 28–40 (1985)

Holm, D., Schneider, D., Coath, C.D.: Age and deformation of early proterozoic quartzites in the southern lake superior region: implications for extent of foreland deformation during final assembly of laurentia. Geology 26(10), 907–910 (1998)

Holme, P., Liljeros, F., Edling, C.R., Kim, B.J.: Network bipartivity. Phys. Rev. E 68(5), 056107 (2003)

Hook, J.: Min-plus algebraic low rank matrix approximation: a new method for revealing structure in networks. arXiv preprint arXiv:1708.06552 (2017)

Hu, D., Li, X., Liu, X., Zhang, S.: The Laplacian energy and Laplacian Estrada index of random multipartite graphs. J. Math. Anal. Appl. 443(2), 675–687 (2016)

Huo, J., Zhao, H.: Dynamical analysis of a fractional sir model with birth and death on heterogeneous complex networks. Phys. A Stat. Mech. Appl. 448, 41–56 (2016)

Ilić, A., Stevanović, D.: The Estrada index of chemical trees. J. Math. Chem. 47(1), 305–314 (2010)

Ilic, A., Zhou, B.: Laplacian Estrada index of trees. arXiv preprint arXiv:1106.3041 (2011)

Ionescu, C., Lopes, A., Copot, D., Machado, J.T., Bates, J.H.: The role of fractional calculus in modeling biological phenomena: a review. Commun. Nonlinear Sci. Numer. Simul. 51, 141–159 (2017)

Ivanciuc, O., Balaban, T.-S., Balaban, A.T.: Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices. J. Math. Chem. 12(1), 309–318 (1993)

Jahanbani, A.: New bounds for the Harary energy and Harary Estrada index of graphs. Mati 1(1), 40–51 (2019)

Joswig, M., Schröter, B.: Parametric shortest-path algorithms via tropical geometry. arXiv preprint arXiv:1904.01082 (2019)

Kamenkovich, I., Berloff, P., Haigh, M., Sun, L., Lu, Y.: Complexity of mesoscale eddy diffusivity in the ocean. Geophys. Res. Lett. 48(5), e2020GL091719 (2021)

Katz, L.: A new status index derived from sociometric analysis. Psychometrika 18(1), 39–43 (1953)

Kazu, R.S., Maldonado, J., Mota, B., Manger, P.R., Herculano-Houzel, S.: Cellular scaling rules for the brain of artiodactyla include a highly folded cortex with few neurons. Front. Neuroanat. 8, 128 (2014)

Keeling, M.J., Eames, K.T.: Networks and epidemic models. J. R. Soc. Interface 2(4), 295–307 (2005)

Ketterson, J.B.: The Physics of Solids. Oxford University Press, Oxford (2016)

Khosravanirad, A.: A lower bound for Laplacian Estrada index of a graph. MATCH Commun. Math. Comput. Chem. 70, 175–180 (2013)

Kim, B.J., Hong, H., Choi, M.: Netons: vibrations of complex networks. J. Phys. A Math. Gen. 36(23), 6329 (2003)

Kim, N., Fuhr, K. N., Schlick, T.: Graph applications to RNA structure and function. In: Biophysics of RNA Folding, pp. 23–51. Springer (2013)

King, A.E., Turner, M.S.: Non-local interactions in collective motion. R. Soc. Open Sci. 8(3), 201536 (2021)

Kiss, I.Z., Miller, J.C., Simon, P.L., et al.: Mathematics of Epidemics on Networks, p. 598. Springer, Cham (2017)

Kloster, K., Gleich, D.F.: Heat kernel based community detection. In: Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1386–1395 (2014)

Kloster, K., Sullivan, B.D., et al.: Walk entropy and walk-regularity. Linear Algebra Appl. 546, 115–121 (2018)

Kok, C.M., Rudin, A.: Relationship between the hydrodynamic radius and the radius of gyration of a polymer in solution. Die Makromol. Chem. Rapid Commun. 2(11), 655–659 (1981)

König, D.: Über graphen und ihre anwendung auf determinantentheorie und mengenlehre. Math. Ann. 77(4), 453–465 (1916)

Kutzelnigg, W.: What i like about Hückel theory. J. Comput. Chem. 28(1), 25–34 (2007)

Lee, C.-H., Tenneti, S., Eun, D.Y.: Transient dynamics of epidemic spreading and its mitigation on large networks. In: Proceedings of the Twentieth ACM International Symposium on Mobile Ad Hoc Networking and Computing, Mobihoc ’19, pp. 191–200. Association for Computing Machinery, New York (2019)

Li, B.: New bounds for Estrada index. MATCH Commun. Math. Comput. Chem. 77, 701–706 (2017)

Li, F., Wei, L., Zhao, H., Hu, F., Ma, X.: On the Estrada index of cactus graphs. Discrete Appl. Math. 203, 94–105 (2016)

Li, J., Guo, J.-M., Shiu, W.C.: The normalized Laplacian Estrada index of a graph. Filomat 28(2), 365–371 (2014)

Li, J., Li, X., Wang, L.: The minimal Estrada index of trees with two maximum degree vertices. MATCH Commun. Math. Comput. Chem. 64, 799–810 (2010)

Li, J., Shiu, W.C., Chan, W.H.: Note on the Laplacian Estrada index of a graph. MATCH Commun. Math. Comput. Chem. 66, 777–784 (2011)

Li, J., Shiu, W.C., Chang, A.: On the Laplacian Estrada index of a graph. Appl. Anal. Discrete Math. 3, 147–156 (2009)

Li, J., Zhang, J.: On the Laplacian Estrada index of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 68(3), 835–842 (2012)

Li, J., Zhang, J.: Characterizing graphs with maximal Laplacian Estrada index. Linear Algebra Appl. 465, 312–324 (2015)

Linderoth, T.R., Horch, S., Lægsgaard, E., Stensgaard, I., Besenbacher, F.: Surface diffusion of pt on pt (110): Arrhenius behavior of long jumps. Phys. Rev. Lett. 78(26), 4978 (1997)

Liu, J.-B., Pan, X.-F., Cao, J.: Some properties on Estrada index of folded hypercubes networks. In: Abstract and Applied Analysis, vol. 2014. Hindawi (2014)

Liu, J.-P., Liu, B.-L.: Bounds of the Estrada index of graphs. Appl. Math. A J. Chin. Univ. 25(3), 325–330 (2010)

Lučić, B., Miličević, A., Nikolić, S., Trinajstić, N.: Harary index-twelve years later. Croat. Chem. Acta 75(4), 847–868 (2002)

Majumdar, C.K., Ghosh, D.K.: On next-nearest-neighbor interaction in linear chain. I. J. Math. Phys. 10(8), 1388–1398 (1969)

Martcheva, M.: An Introduction to Mathematical Epidemiology, vol. 61. Springer, New York (2015)

Mei, W., Mohagheghi, S., Zampieri, S., Bullo, F.: On the dynamics of deterministic epidemic propagation over networks. Annu. Rev. Control 44, 116–128 (2017)

Miyamoto, S., Shimono, K.: Molecular modeling to estimate the diffusion coefficients of drugs and other small molecules. Molecules 25(22), 5340 (2020)

Mugnolo, D.: Dynamical systems associated with adjacency matrices. Discrete Contin. Dyn. Syst. B 23(5), 1945–1973 (2018)

Nag, B., Chattopadhyay, D.: Negative electron diffusivity in high electric fields. Solid-State Electron. 21(1), 303–307 (1978)

Nagatani, T.: Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction. Phys. Rev. E 60(6), 6395 (1999)

Nasiri, R., Ellahi, H., Gholami, A., Fath-Tabar, G., Ashrafi, A.: Resolvent Estrada and signless Laplacian Estrada indices of graphs. MATCH Commun. Math. Comput. Chem. 77(1), 157–176 (2017)

Nguyen, A.Q., Anh, N.T., Nguyên, T.A.: Frontier Orbitals: A Practical Manual. Wiley, New York (2007)

Ninomiya, H., Tanaka, Y., Yamamoto, H.: Reaction, diffusion and non-local interaction. J. Math. Biol. 75(5), 1203–1233 (2017)

Onsager, L.: Statistical hydrodynamics. Il Nuovo Cimento (1943–1954) 6(2), 279–287 (1949)

Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)

Pisanski, T., Randić, M.: Use of the Szeged index and the revised Szeged index for measuring network bipartivity. Discrete Appl. Math. 158(17), 1936–1944 (2010)

Plavšić, D., Nikolić, S., Trinajstić, N., Mihalić, Z.: On the Harary index for the characterization of chemical graphs. J. Math. Chem. 12(1), 235–250 (1993)

Pope, M., Swenberg, C.E., et al.: Electronic Processes in Organic Crystals and Polymers, vol. 39. Oxford University Press on Demand (1999)

Popolizio, M.: On the matrix Mittag–Leffler function: theoretical properties and numerical computation. Mathematics 7(12), 1140 (2019)

Powell, B.J.: An introduction to effective low-energy hamiltonians in condensed matter physics and chemistry. arXiv preprint arXiv:0906.1640 (2009)

Rad, N.J., Jahanbani, A., Hasni, R.: Pentacyclic graphs with maximal Estrada index. Ars Combin. 133, 133–145 (2017)

Rad, N.J., Jahanbani, A., Mojdeh, D.A.: Tetracyclic graphs with maximal Estrada index. Discrete Math. Algorithms Appl. 9(03), 1750041 (2017)

Rashid, M.A., Ahmad, S., Siddiqui, M.K., Jahanbani, A., Sheikholeslami, S., Shao, Z.: New bounds for the Estrada index of phenylenes. Polycyclic Aromatic Compounds, pp. 1–17 (2020)

Robbiano, M., Jimenez, R., Medina, L.: The energy and an approximation to Estrada index of some trees. Match 61(2), 369 (2009)

Rodríguez, J.: A note on new bounds for the Estrada index. Linear Algebra Appl. 580, 121–127 (2019)

Rodríguez, J., Aguayo, J.L., Carmona, J.R., Jahanbani, A.: A note lower bounds for the Estrada index. Discrete Math. 344(4), 112303 (2021)

Saloff-Coste, L.: The heat kernel and its estimates. In: Probabilistic Approach to Geometry, pp. 405–436. Mathematical Society of Japan (2010)

Sardar, T., Rana, S., Bhattacharya, S., Al-Khaled, K., Chattopadhyay, J.: A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector. Math. Biosci. 263, 18–36 (2015)

Sasaki, K.-I., Shimomura, Y., Takane, Y., Wakabayashi, K.: Hamiltonian decomposition for bulk and surface states. Phys. Rev. Lett. 102(14), 146806 (2009)

Schunack, M., Linderoth, T.R., Rosei, F., Lægsgaard, E., Stensgaard, I., Besenbacher, F.: Long jumps in the surface diffusion of large molecules. Phys. Rev. Lett. 88(15), 156102 (2002)

Senft, D.C., Ehrlich, G.: Long jumps in surface diffusion: one-dimensional migration of isolated adatoms. Phys. Rev. Lett. 74(2), 294 (1995)

Shang, Y.: The Estrada index of random graphs. Sci. Magna 7, 79–81 (2011)

Shang, Y.: Perturbation results for the Estrada index in weighted networks. J. Phys. A Math. Theor. 44(7), 075003 (2011)

Shang, Y.: Lower bounds for the Estrada index. Electron. J. Linear Algebra 23, 664–668 (2012)

Shang, Y.: On the Estrada index of Ramanujan graphs. J. Combin. Inf. Syst. Sci. 37(1), 69 (2012)

Shang, Y.: Estrada index of general weighted graphs. Bull. Aust. Math. Soc. 88(1), 106–112 (2013)

Shang, Y.: Lower bounds for the Estrada index using mixing time and Laplacian spectrum. Rocky Mt. J. Math. 43(6), 2009–2016 (2013)

Shang, Y.: Estimating the distance Estrada index. arXiv preprint arXiv:1407.5248 (2014)

Shang, Y.: More on the normalized Laplacian Estrada index. Appl. Anal. Discrete Math. 8, 346–357 (2014)

Shang, Y.: Bounds of distance Estrada index of graphs. arXiv preprint arXiv:1511.06132 (2015)

Shang, Y.: Distance Estrada index of random graphs. Linear Multilinear Algebra 63(3), 466–471 (2015)

Shang, Y.: Estrada index of random bipartite graphs. Symmetry 7(4), 2195–2205 (2015)

Shang, Y.: Laplacian Estrada and normalized Laplacian Estrada indices of evolving graphs. PLoS One 10(3), e0123426 (2015)

Shang, Y.: On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs. Open Math. 14(1), 641–648 (2016)

Shang, Y.: Further results on distance Estrada index of random graphs. Bull. Malays. Math. Sci. Soc. 41(2), 537–544 (2018)

Shang, Y.: Lower bounds for gaussian Estrada index of graphs. Symmetry 10(8), 325 (2018)

Sladek, V., Harada, R., Shigeta, Y.: Protein dynamics and the folding degree. J. Chem. Inf. Model. 60(3), 1559–1567 (2020)

Sladek, V., Tokiwa, H., Shimano, H., Shigeta, Y.: Protein residue networks from energetic and geometric data: are they identical? J. Chem. Theory Comput. 14(12), 6623–6631 (2018)

Spielman, D.: Spectral graph theory. Combinatorial Scientific Computing, chapter 18. CRC Press, Boca Raton (2012)

Stanić, Z.: Inequalities for Graph Eigenvalues, vol. 423. Cambridge University Press, Cambridge (2015)

Trinajstić, N.: Hückel theory and topology. In: Semiempirical Methods of Electronic Structure Calculation, pp. 1–28. Springer, New York (1977)

Trinajstić, N.: Graph theory and molecular orbitals, pp. 235–275. Introduction and Fundamentals, Mathematical Chemistry Series, Chemical Graph Theory (1991)

Trinajstic, N.: Chemical Graph Theory. Routledge, New York (2018)

Tsuji, Y., Estrada, E.: Influence of long-range interactions on quantum interference in molecular conduction. A tight-binding (Hückel) approach. J. Chem. Phys. 150(20), 204123 (2019)

Tsuji, Y., Estrada, E., Movassagh, R., Hoffmann, R.: Quantum interference, graphs, walks, and polynomials. Chem. Rev. 118(10), 4887–4911 (2018)

Van Der Hofstad, R.: Random Graphs and Complex Networks, vol. 43. Cambridge University Press, Cambridge (2017)

van Lint, J.H., Seidel, J.J.: Equilateral point sets in elliptic geometry. Indag. Math. 28(3), 335–348 (1966)

Verde-Star, L.: Functions of matrices. Linear Algebra Appl. 406, 285–300 (2005)

Von Collatz, L., Sinogowitz, U.: Spektren endlicher grafen. In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 21, pp. 63–77. Springer, New York (1957)

Von Mering, C., Krause, R., Snel, B., Cornell, M., Oliver, S.G., Fields, S., Bork, P.: Comparative assessment of large-scale data sets of protein–protein interactions. Nature 417(6887), 399–403 (2002)

Vu, B.D., Tong, B.T. et al.: Dynamics of negative diffusivity and uphill diffusion in ternary and single systems. In: EPJ Web of Conferences, vol. 206, p. 09015. EDP Sciences (2019)

Wang, F., Stuart, J.: Instability of Hagen–Poiseuille flow for non-axisymmetric mode. Appl. Math. Mech. 10(5), 407–417 (1989)

Wang, K., Pan, X., Ning, W.: On the signless Laplacian Estrada index of cacti. Discrete Appl. Math. 254, 189–195 (2019)

Wang, L., Fan, Y.-Z., Wang, Y.: Maximum Estrada index of bicyclic graphs. Discrete Appl. Math. 180, 194–199 (2015)

Wang, L.-W., Zunger, A.: Solving Schrödinger’s equation around a desired energy: application to silicon quantum dots. J. Chem. Phys. 100(3), 2394–2397 (1994)

Wang, W.L., Yazyev, O.V., Meng, S., Kaxiras, E.: Topological frustration in graphene nanoflakes: magnetic order and spin logic devices. Phys. Rev. Lett. 102(15), 157201 (2009)

Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)

White, J.G., Southgate, E., Thomson, J.N., Brenner, S., et al.: The structure of the nervous system of the nematode Caenorhabditis elegans. Philos. Trans. R. Soc. Lond. B Biol. Sci. 314(1165), 1–340 (1986)

Wigner, E.P.: The unreasonable effectiveness of mathematics in the natural sciences. In: Mathematics and Science, pp. 291–306. World Scientific (1990)

Woodward, R.B., Hoffmann, R.: Stereochemistry of electrocyclic reactions. J. Am. Chem. Soc. 87(2), 395–397 (1965)

Wrigley, J.D., Twigg, M.E., Ehrlich, G.: Lattice walks by long jumps. J. Chem. Phys. 93(4), 2885–2902 (1990)

Xiao, B., Hancock, E.R., Wilson, R.C.: Graph characteristics from the heat kernel trace. Pattern Recognit. 42(11), 2589–2606 (2009)

Yamamoto, K., Yoshikura, H.: Relation between genomic and capsid structures in RNA viruses. Nucleic Acids Res. 14(1), 389–396 (1986)

Yang, Y., Sun, L., Bu, C.: A note on some bounds of the Estrada index of graphs. Adv. Math. Phys. 2020, 3972789 (2020)

Yates, K.: Hückel Molecular Orbital Theory. Elsevier, Amsterdam (2012)

Yu, C., Guan, J., Chen, K., Bae, S.C., Granick, S.: Single-molecule observation of long jumps in polymer adsorption. ACS Nano 7(11), 9735–9742 (2013)

Zaslavsky, T.: A mathematical bibliography of signed and gain graphs and allied areas. Electron. J. Combin. (2012)

Zhang, H., Wang, Y.: On the maximum Laplacian Estrada index of graphs. Linear Algebra Appl. 492, 68–77 (2016)

Zhang, H., Wang, Y.: On the maximum Laplacian Estrada index of graphs. Linear Algebra Appl. 492, 68–77 (2016)

Zhang, J., Zhou, B., Li, J.: On Estrada index of trees. Linear Algebra Appl. 434(1), 215–223 (2011)

Zhao, H., Jia, Y.: On the Estrada index of bipartite graph. Match 61(2), 495 (2009)

Zhao, T., Qiu, D.: One-pot synthesis of highly folded microparticles by suspension polymerization. Langmuir 27(21), 12771–12774 (2011)

Zhou, B.: On Estrada index. MATCH Commun. Math. Comput. Chem. 60, 485–492 (2008)

Zhou, B.: On sum of powers of Laplacian eigenvalues and Laplacian Estrada index of graphs. arXiv preprint arXiv:1102.1144 (2011)

Zhou, B., Gutman, I.: More on the Laplacian Estrada index. Appl. Anal. Discrete Math. 3(2), 371–378 (2009)

Zhou, B., Trinajstic, N.: Estrada index of bipartite graphs. Int. J. Chem. Model. 1(3/4), 387–394 (2008)

Zhu, B.-X.: On the Laplacian Estrada index of graphs. MATCH Commun. Math. Comput. Chem. 66(3), 769–776 (2011)

Zhu, Z., Tan, L., Qiu, Z.: Tricyclic graph with maximal Estrada index. Discrete Appl. Math. 162, 364–372 (2014)