doi: 10.3934/cpaa.2021095

On the hot spots of quantum graphs

1. 

Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016 Lisboa, Portugal

2. 

Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden

* Corresponding author

Received  October 2020 Revised  April 2021 Published  June 2021

We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity–Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

Citation: James B. Kennedy, Jonathan Rohleder. On the hot spots of quantum graphs. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021095
References:
[1]

R. AdamiE. Serra and P. Tilli, Negative energy ground states for the $L^2$-critical NLSE on metric graphs, Commun. Math. Phys., 352 (2017), 387-406.  doi: 10.1007/s00220-016-2797-2.  Google Scholar

[2]

R. AdamiE. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal., 271 (2016), 201-223.  doi: 10.1016/j.jfa.2016.04.004.  Google Scholar

[3]

R. AdamiE. Serra and P. Tilli, NLS ground states on graphs, Calc. Var. Partial Differ. Equ., 54 (2015), 743-761.  doi: 10.1007/s00526-014-0804-z.  Google Scholar

[4]

M. AizenmanH. SchanzU. Smilansky and S. Warzel, Edge switching transformations of quantum graphs, Acta Phys. Polon. A, 132 (2017), 1699-1703.   Google Scholar

[5]

L. Alon, Quantum graphs–Generic eigenfunctions and their nodal count and Neumann count statistics, Ph.D thesis, Technion, Israel, arXiv: 2010.03004. Google Scholar

[6]

L. Alon and R. Band, Neumann domains on quantum graphs, arXiv: 1911.12435. Google Scholar

[7]

L. AlonR. Band and G. Berkolaiko, Nodal statistics on quantum graphs, Commun. Math. Phys., 362 (2018), 909-948.  doi: 10.1007/s00220-018-3111-2.  Google Scholar

[8]

L. Alon, R. Band, M. Bersudsky and S. Egger, Neumann Domains on Graphs and Manifolds, arXiv: 1805.07612. Google Scholar

[9]

S Ariturk, Eigenvalue estimates on quantum graphs, arXiv: 1609.07471. Google Scholar

[10]

R. Band, The nodal count $\{0, 1, 2, 3, \ldots\}$ implies the graph is a tree, Philos. Trans. R. Soc. Lond. A, 372 (2014), (24pp). doi: 10.1098/rsta.2012.0504.  Google Scholar

[11]

R. BandG. BerkolaikoH. Raz and U. Smilansky, The number of nodal domains of graphs as a stability index of graph partitions, Commun. Math. Phys., 311 (2012), 815-838.  doi: 10.1007/s00220-011-1384-9.  Google Scholar

[12]

R. BandG. Berkolaiko and U. Smilansky, Dynamics of nodal points and the nodal count of a family of quantum graphs, Ann. Henri Poincaré, 13 (2012), 145-184.  doi: 10.1007/s00023-011-0124-1.  Google Scholar

[13]

R. Band and D. Fajman, Topological properties of Neumann domains, Ann. Henri Poincaré, 17 (2016), 2379-2407.  doi: 10.1007/s00023-016-0468-7.  Google Scholar

[14]

R. Band and G. Lévy, Quantum graphs which optimize the spectral gap, Ann. Henri Poincaré, 18 (2017), 3269-3323.  doi: 10.1007/s00023-017-0601-2.  Google Scholar

[15]

R. Bañuelos and K. Burdzy, On the "hot spots" conjecture of J. Rauch, J. Funct. Anal., 164 (1999), 1-33.  doi: 10.1006/jfan.1999.3397.  Google Scholar

[16]

G. BerkolaikoJ. B. KennedyP. Kurasov and D. Mugnolo, Surgery principles for the spectral analysis of quantum graphs, Trans. Amer. Math. Soc., 372 (2019), 5153-5197.  doi: 10.1090/tran/7864.  Google Scholar

[17]

G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Edge connectivity and the spectral gap of combinatorial and quantum graphs, J. Phys. A: Math. Theor., 50 (2017), 365201 (29pp). doi: 10.1088/1751-8121/aa8125.  Google Scholar

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G. Berkolaiko and P. Kuchment, Introduction to quantum graphs. Math. Surveys and Monographs vol. 186, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.  Google Scholar

[19]

G. Berkolaiko and P. Kuchment, Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths, Spectral Geometry, 117–137, Proc. Sympos. Pure Math., vol.84, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/pspum/084/1352.  Google Scholar

[20]

G. BerkolaikoY. Latushkin and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, Adv. Math., 352 (2019), 632-669.  doi: 10.1016/j.aim.2019.06.017.  Google Scholar

[21]

G. Berkolaiko and W. Liu, Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph, J. Math. Anal. Appl., 445 (2017), 803-818.  doi: 10.1016/j.jmaa.2016.07.026.  Google Scholar

[22]

D. Borthwick, L. Corsi and K. Jones, Sharp diameter bound on the spectral gap for quantum graphs, arXiv: 1905.03071. Google Scholar

[23]

K. Burdzy and W. Werner, A counterexample to the "hot spots" conjecture, Ann. Math., 149 (1999), 309-317.  doi: 10.2307/121027.  Google Scholar

[24]

C. Cacciapuoti, Scale invariant effective Hamiltonians for a graph with a small compact core, Symmetry, 11 (2019), 359. Google Scholar

[25]

C. CacciapuotiD. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303.  doi: 10.1088/1361-6544/aa7cc3.  Google Scholar

[26]

J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian. in Problems in analysis, Princeton Univ. Press, Princeton, N. J., 1970, 195-199.  Google Scholar

[27]

M. K. Chung, S. Seo, N. Adluru and H. K. Vorperian, Hot Spots Conjecture and Its Application to Modeling Tubular Structures. In K. Suzuki, F. Wang, D. Shen and P. Yan (eds), Machine Learning in Medical Imaging, Lecture Notes in Computer Science, vol. 7009, Springer, Berlin–Heidelberg, 2011,225–232. Google Scholar

[28]

S. Dovetta and L. Tentarelli, $L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features, Calc. Var. Partial Differ. Equ., 58 (2019), 26 pp. doi: 10.1007/s00526-019-1565-5.  Google Scholar

[29]

L. C. Evans, The Fiedler Rose: On the extreme points of the Fiedler vector, arXiv: 1112.6323. Google Scholar

[30]

L. Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble), 55 (2005), 199-211.   Google Scholar

[31]

L. Friedlander, Genericity of simple eigenvalues for a metric graph, Israel J. Math., 146 (2005), 149-156.  doi: 10.1007/BF02773531.  Google Scholar

[32]

H. Gernandt and J. P. Pade, Schur reduction of trees and extremal entries of the Fiedler vector, Linear Algebra Appl., 570 (2019), 93-122.  doi: 10.1016/j.laa.2019.02.008.  Google Scholar

[33]

S. Gnutzmann, U. Smilansky and J. Weber, Nodal counting on quantum graphs, Special section on quantum graphs, Waves Random Media, 14 (2004), S61–S73. doi: 10.1088/0959-7174/14/1/011.  Google Scholar

[34]

E. M. Harrell II and A. V. Maltsev, Localization and landscape functions on quantum graphs, arXiv: 1803.01186. doi: 10.1090/tran/7908.  Google Scholar

[35]

E. M. Harrell II and A. V. Maltsev, On Agmon metrics and exponential localization for quantum graphs, Commun. Math. Phys., 359 (2018), 429-448.  doi: 10.1007/s00220-018-3124-x.  Google Scholar

[36]

M. Hofmann, An existence theory for nonlinear equations on metric graphs via energy methods, arXiv: 1909.07856. Google Scholar

[37]

M. Hofmann, J. B. Kennedy, D. Mugnolo and M. Plümer, Asymptotics and estimates for spectral minimal partitions of metric graphs, arXiv: 2007.01412. Google Scholar

[38]

C. Judge and S. Mondal, Euclidean triangles have no hot spots, Ann. Math., 191 (2020), 167-211.  doi: 10.4007/annals.2020.191.1.3.  Google Scholar

[39]

A. Kairzhan, D. E. Pelinovsky and R. H. Goodman, Drift of spectrally stable shifted states on star graphs, SIAM J. Appl. Dyn. Syst., 18 (2019), 1723–1755. doi: 10.1137/19M1246146.  Google Scholar

[40]

G. Karreskog, P. Kurasov and I. Trygg Kupersmidt, Schrödinger operators on graphs: symmetrization and Eulerian cycles, Proc. Amer. Math. Soc., 144 (2016) 1197–1207. doi: 10.1090/proc12784.  Google Scholar

[41]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

[42]

J. B. Kennedy, P. Kurasov, C. Léna and D. Mugnolo, A theory of spectral partitions of metric graphs, Calc. Var. Partial Differ. Equ., 60 (2021), 63 pp. doi: 10.1007/s00526-021-01966-y.  Google Scholar

[43]

J. B. KennedyP. KurasovG. Malenová and D. Mugnolo, On the spectral gap of a quantum graph, Ann. Henri Poincaré, 17 (2016), 2439-2473.  doi: 10.1007/s00023-016-0460-2.  Google Scholar

[44]

J. B. Kennedy and J. Rohleder, On the hot spots of quantum trees, Proc. Appl. Math. Mech., 18 (2018), e201800122. Google Scholar

[45]

D. Krejčiřík and M. Tušek, Location of hot spots in thin curved strips, J. Differ. Equ., 266 (2019), 2953-2977.  doi: 10.1016/j.jde.2018.08.053.  Google Scholar

[46]

P. Kurasov, G. Malenová and S. Naboko, Spectral gap for quantum graphs and their connectivity, J. Phys. A, 46 (2013), 275309. doi: 10.1088/1751-8113/46/27/275309.  Google Scholar

[47]

P. Kurasov and S. Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory, 4 (2014), 211-219.  doi: 10.4171/JST/67.  Google Scholar

[48]

C. LangeS. LiuN. Peyerimhoff and O. Post, Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians, Calc. Var. Partial Differ. Equ., 54 (2015), 4165-4196.  doi: 10.1007/s00526-015-0935-x.  Google Scholar

[49]

R. Lederman and S. Steinerberger, Extreme values of the Fiedler vector on trees, arXiv: 1912.08327. Google Scholar

[50]

J. R. Lee, S. O. Gharan and L. Trevisan, Multiway spectral partitioning and higher-order Cheeger inequalities, J. ACM, 61 (2014), 30 pp. doi: 10.1145/2665063.  Google Scholar

[51]

J. Lefèvre, Fiedler vectors and elongation of graphs: a threshold phenomenon on a particular class of trees, arXiv: 1302.1266. Google Scholar

[52]

J. Rohleder, Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc., 145 (2017), 2119-2129.  doi: 10.1090/proc/13403.  Google Scholar

[53]

J. Rohleder and C. Seifert, Spectral monotonicity for Schrödinger operators on metric graphs, Oper. Theory Adv. Appl., 281 (2020), 291-310.   Google Scholar

[54]

B. Siudeja, Hot spots conjecture for a class of acute triangles, Math. Z., 280 (2015), 783-806.  doi: 10.1007/s00209-015-1448-1.  Google Scholar

[55]

S. Steinerberger, Hot Spots in Convex Domains are in the Tips (up to an Inradius), Commun. Partial Differ. Equ., 45 (2020), 641-654.  doi: 10.1080/03605302.2020.1750427.  Google Scholar

show all references

References:
[1]

R. AdamiE. Serra and P. Tilli, Negative energy ground states for the $L^2$-critical NLSE on metric graphs, Commun. Math. Phys., 352 (2017), 387-406.  doi: 10.1007/s00220-016-2797-2.  Google Scholar

[2]

R. AdamiE. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal., 271 (2016), 201-223.  doi: 10.1016/j.jfa.2016.04.004.  Google Scholar

[3]

R. AdamiE. Serra and P. Tilli, NLS ground states on graphs, Calc. Var. Partial Differ. Equ., 54 (2015), 743-761.  doi: 10.1007/s00526-014-0804-z.  Google Scholar

[4]

M. AizenmanH. SchanzU. Smilansky and S. Warzel, Edge switching transformations of quantum graphs, Acta Phys. Polon. A, 132 (2017), 1699-1703.   Google Scholar

[5]

L. Alon, Quantum graphs–Generic eigenfunctions and their nodal count and Neumann count statistics, Ph.D thesis, Technion, Israel, arXiv: 2010.03004. Google Scholar

[6]

L. Alon and R. Band, Neumann domains on quantum graphs, arXiv: 1911.12435. Google Scholar

[7]

L. AlonR. Band and G. Berkolaiko, Nodal statistics on quantum graphs, Commun. Math. Phys., 362 (2018), 909-948.  doi: 10.1007/s00220-018-3111-2.  Google Scholar

[8]

L. Alon, R. Band, M. Bersudsky and S. Egger, Neumann Domains on Graphs and Manifolds, arXiv: 1805.07612. Google Scholar

[9]

S Ariturk, Eigenvalue estimates on quantum graphs, arXiv: 1609.07471. Google Scholar

[10]

R. Band, The nodal count $\{0, 1, 2, 3, \ldots\}$ implies the graph is a tree, Philos. Trans. R. Soc. Lond. A, 372 (2014), (24pp). doi: 10.1098/rsta.2012.0504.  Google Scholar

[11]

R. BandG. BerkolaikoH. Raz and U. Smilansky, The number of nodal domains of graphs as a stability index of graph partitions, Commun. Math. Phys., 311 (2012), 815-838.  doi: 10.1007/s00220-011-1384-9.  Google Scholar

[12]

R. BandG. Berkolaiko and U. Smilansky, Dynamics of nodal points and the nodal count of a family of quantum graphs, Ann. Henri Poincaré, 13 (2012), 145-184.  doi: 10.1007/s00023-011-0124-1.  Google Scholar

[13]

R. Band and D. Fajman, Topological properties of Neumann domains, Ann. Henri Poincaré, 17 (2016), 2379-2407.  doi: 10.1007/s00023-016-0468-7.  Google Scholar

[14]

R. Band and G. Lévy, Quantum graphs which optimize the spectral gap, Ann. Henri Poincaré, 18 (2017), 3269-3323.  doi: 10.1007/s00023-017-0601-2.  Google Scholar

[15]

R. Bañuelos and K. Burdzy, On the "hot spots" conjecture of J. Rauch, J. Funct. Anal., 164 (1999), 1-33.  doi: 10.1006/jfan.1999.3397.  Google Scholar

[16]

G. BerkolaikoJ. B. KennedyP. Kurasov and D. Mugnolo, Surgery principles for the spectral analysis of quantum graphs, Trans. Amer. Math. Soc., 372 (2019), 5153-5197.  doi: 10.1090/tran/7864.  Google Scholar

[17]

G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Edge connectivity and the spectral gap of combinatorial and quantum graphs, J. Phys. A: Math. Theor., 50 (2017), 365201 (29pp). doi: 10.1088/1751-8121/aa8125.  Google Scholar

[18]

G. Berkolaiko and P. Kuchment, Introduction to quantum graphs. Math. Surveys and Monographs vol. 186, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.  Google Scholar

[19]

G. Berkolaiko and P. Kuchment, Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths, Spectral Geometry, 117–137, Proc. Sympos. Pure Math., vol.84, Amer. Math. Soc., Providence, RI, 2012. doi: 10.1090/pspum/084/1352.  Google Scholar

[20]

G. BerkolaikoY. Latushkin and S. Sukhtaiev, Limits of quantum graph operators with shrinking edges, Adv. Math., 352 (2019), 632-669.  doi: 10.1016/j.aim.2019.06.017.  Google Scholar

[21]

G. Berkolaiko and W. Liu, Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph, J. Math. Anal. Appl., 445 (2017), 803-818.  doi: 10.1016/j.jmaa.2016.07.026.  Google Scholar

[22]

D. Borthwick, L. Corsi and K. Jones, Sharp diameter bound on the spectral gap for quantum graphs, arXiv: 1905.03071. Google Scholar

[23]

K. Burdzy and W. Werner, A counterexample to the "hot spots" conjecture, Ann. Math., 149 (1999), 309-317.  doi: 10.2307/121027.  Google Scholar

[24]

C. Cacciapuoti, Scale invariant effective Hamiltonians for a graph with a small compact core, Symmetry, 11 (2019), 359. Google Scholar

[25]

C. CacciapuotiD. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303.  doi: 10.1088/1361-6544/aa7cc3.  Google Scholar

[26]

J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian. in Problems in analysis, Princeton Univ. Press, Princeton, N. J., 1970, 195-199.  Google Scholar

[27]

M. K. Chung, S. Seo, N. Adluru and H. K. Vorperian, Hot Spots Conjecture and Its Application to Modeling Tubular Structures. In K. Suzuki, F. Wang, D. Shen and P. Yan (eds), Machine Learning in Medical Imaging, Lecture Notes in Computer Science, vol. 7009, Springer, Berlin–Heidelberg, 2011,225–232. Google Scholar

[28]

S. Dovetta and L. Tentarelli, $L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features, Calc. Var. Partial Differ. Equ., 58 (2019), 26 pp. doi: 10.1007/s00526-019-1565-5.  Google Scholar

[29]

L. C. Evans, The Fiedler Rose: On the extreme points of the Fiedler vector, arXiv: 1112.6323. Google Scholar

[30]

L. Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier (Grenoble), 55 (2005), 199-211.   Google Scholar

[31]

L. Friedlander, Genericity of simple eigenvalues for a metric graph, Israel J. Math., 146 (2005), 149-156.  doi: 10.1007/BF02773531.  Google Scholar

[32]

H. Gernandt and J. P. Pade, Schur reduction of trees and extremal entries of the Fiedler vector, Linear Algebra Appl., 570 (2019), 93-122.  doi: 10.1016/j.laa.2019.02.008.  Google Scholar

[33]

S. Gnutzmann, U. Smilansky and J. Weber, Nodal counting on quantum graphs, Special section on quantum graphs, Waves Random Media, 14 (2004), S61–S73. doi: 10.1088/0959-7174/14/1/011.  Google Scholar

[34]

E. M. Harrell II and A. V. Maltsev, Localization and landscape functions on quantum graphs, arXiv: 1803.01186. doi: 10.1090/tran/7908.  Google Scholar

[35]

E. M. Harrell II and A. V. Maltsev, On Agmon metrics and exponential localization for quantum graphs, Commun. Math. Phys., 359 (2018), 429-448.  doi: 10.1007/s00220-018-3124-x.  Google Scholar

[36]

M. Hofmann, An existence theory for nonlinear equations on metric graphs via energy methods, arXiv: 1909.07856. Google Scholar

[37]

M. Hofmann, J. B. Kennedy, D. Mugnolo and M. Plümer, Asymptotics and estimates for spectral minimal partitions of metric graphs, arXiv: 2007.01412. Google Scholar

[38]

C. Judge and S. Mondal, Euclidean triangles have no hot spots, Ann. Math., 191 (2020), 167-211.  doi: 10.4007/annals.2020.191.1.3.  Google Scholar

[39]

A. Kairzhan, D. E. Pelinovsky and R. H. Goodman, Drift of spectrally stable shifted states on star graphs, SIAM J. Appl. Dyn. Syst., 18 (2019), 1723–1755. doi: 10.1137/19M1246146.  Google Scholar

[40]

G. Karreskog, P. Kurasov and I. Trygg Kupersmidt, Schrödinger operators on graphs: symmetrization and Eulerian cycles, Proc. Amer. Math. Soc., 144 (2016) 1197–1207. doi: 10.1090/proc12784.  Google Scholar

[41]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.  Google Scholar

[42]

J. B. Kennedy, P. Kurasov, C. Léna and D. Mugnolo, A theory of spectral partitions of metric graphs, Calc. Var. Partial Differ. Equ., 60 (2021), 63 pp. doi: 10.1007/s00526-021-01966-y.  Google Scholar

[43]

J. B. KennedyP. KurasovG. Malenová and D. Mugnolo, On the spectral gap of a quantum graph, Ann. Henri Poincaré, 17 (2016), 2439-2473.  doi: 10.1007/s00023-016-0460-2.  Google Scholar

[44]

J. B. Kennedy and J. Rohleder, On the hot spots of quantum trees, Proc. Appl. Math. Mech., 18 (2018), e201800122. Google Scholar

[45]

D. Krejčiřík and M. Tušek, Location of hot spots in thin curved strips, J. Differ. Equ., 266 (2019), 2953-2977.  doi: 10.1016/j.jde.2018.08.053.  Google Scholar

[46]

P. Kurasov, G. Malenová and S. Naboko, Spectral gap for quantum graphs and their connectivity, J. Phys. A, 46 (2013), 275309. doi: 10.1088/1751-8113/46/27/275309.  Google Scholar

[47]

P. Kurasov and S. Naboko, Rayleigh estimates for differential operators on graphs, J. Spectr. Theory, 4 (2014), 211-219.  doi: 10.4171/JST/67.  Google Scholar

[48]

C. LangeS. LiuN. Peyerimhoff and O. Post, Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians, Calc. Var. Partial Differ. Equ., 54 (2015), 4165-4196.  doi: 10.1007/s00526-015-0935-x.  Google Scholar

[49]

R. Lederman and S. Steinerberger, Extreme values of the Fiedler vector on trees, arXiv: 1912.08327. Google Scholar

[50]

J. R. Lee, S. O. Gharan and L. Trevisan, Multiway spectral partitioning and higher-order Cheeger inequalities, J. ACM, 61 (2014), 30 pp. doi: 10.1145/2665063.  Google Scholar

[51]

J. Lefèvre, Fiedler vectors and elongation of graphs: a threshold phenomenon on a particular class of trees, arXiv: 1302.1266. Google Scholar

[52]

J. Rohleder, Eigenvalue estimates for the Laplacian on a metric tree, Proc. Amer. Math. Soc., 145 (2017), 2119-2129.  doi: 10.1090/proc/13403.  Google Scholar

[53]

J. Rohleder and C. Seifert, Spectral monotonicity for Schrödinger operators on metric graphs, Oper. Theory Adv. Appl., 281 (2020), 291-310.   Google Scholar

[54]

B. Siudeja, Hot spots conjecture for a class of acute triangles, Math. Z., 280 (2015), 783-806.  doi: 10.1007/s00209-015-1448-1.  Google Scholar

[55]

S. Steinerberger, Hot Spots in Convex Domains are in the Tips (up to an Inradius), Commun. Partial Differ. Equ., 45 (2020), 641-654.  doi: 10.1080/03605302.2020.1750427.  Google Scholar

Figure 3.1.  Left: a path graph with the hot spots marked in grey. Right: a cycle graph; here every point is a hot spot
Figure 3.2.  A pumpkin and a star graph. The set $ M (\Gamma) $ in the equilateral case is marked in grey
Figure 3.3.  A flower graph and a complete graph. The hot spots for the equilateral case are marked in grey
Figure 3.4.  A lasso graph with its hot spots in grey
Figure 4.1.  The perturbed "figure-8" graph of Example 4.7 and its hot spots in grey
Figure 4.2.  The perturbed path graph of Example 4.8 and its hot spots in grey
Figure 5.1.  The complete graph admits an eigenfunction for $ \mu_2 $ whose maximum is at $ v_0 $ and minimum is achieved at the other $ v_k $, with no other critical points
Figure 6.1.  The star graph $ \Gamma^* $, the intermediate tree $ \Gamma^\top $, and the final, symmetric tree $ \Gamma $ in the case $ m = 5 $
Figure 7.1.  The discrete graph $ {{\mathcal G}} $, a "pumpkin on a stick"
Figure 7.1 together with their hot spots (grey)">Figure 7.2.  Metric graph incarnations of the discrete graph $ {{\mathcal G}} $ from Figure 7.1 together with their hot spots (grey)
Figure 8.1.  A graph for which we expect that $ M = \{v_-, v_2, v_3\} $ is possible even if the edge lengths are incommensurable
Figure 8.2.  A candidate graph for Conjecture 8.9. Candidate locations for the hot spots are marked in grey (the precise location will depend on the respective edge lengths and the "thickness" of the pumpkins, and could be within the pumpkins)
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