# Supremizers whose spectral gap is a simple eigenvalue (section 3.5)

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## Supremizers of tree graphs (section 3.4)

Theorem 3.2.2. Let G be a tree graph with El 2 leaves. Then the unique supremizer of G is the equilateral star with El edges, whose spectral gap is 2El. In particular, the uniqueness implies that
this supremizer is a maximizer if and only if G is a star graph. Theorem 3.2.2 completely solves the optimization problem for tree graphs. While writing this paper, we became aware of the recent work, [76], which solves the maximization problem for trees (theorem 3.2 there). In the course of doing so, that work provides the upper bound 2E on the spectral gap of trees 3. Our proof is close in spirit to that of theorem 3.4 in [76]. Yet, thanks to a basic geometric observation (Lemma 3.4.2 here), the better bound 2El is obtained 4.
Theorem 3.2.2 allows to deduce the following.
Corollary 3.2.3. Let G be a non-tree graph. Then its supremizer is not a tree graph.

### Supremizers whose spectral gap is a simple eigenvalue (section 3.5)

Whenever the spectral gap is a simple eigenvalue, it is differentiable with respect to edge lengths, which allows to search for local maximizers. There are indeed examples for critical values (not just maximizers) of the spectral gap, which we demonstrate in Proposition 3.5.8. If such local critical point is actually a supremizer it is possible to prove the following.
Theorem 3.2.4. Let G be a discrete graph and let l 2 LG. Assume that 􀀀 (G; l) is a supremizer of G and that the spectral gap k1 (􀀀(G; l)) is a simple eigenvalue. Then 􀀀 (G; l) is not a unique supremizer. There exists a choice of lengths l 2 LG such that 􀀀 (G; l) is an equilateral mandarin and k1 (􀀀 (G; l)) = k1 (􀀀 (G; l)) .

#### Supremizers of vertex connectivity one (sections 3.6, 3.7, 3.8)

Next, we describe a bottom to top construction which allows to find out a supremizer of a graph by knowing the supremizers of two of its subgraphs. This is possible for graphs of vertex connectivity one. In order to state the result, the following criteria are introduced.
Definition 3.2.5. 1. A Neumann graph 􀀀 obeys the Dirichlet criterion with respect to its vertex v if imposing Dirichlet vertex condition at v does not change the value of k1 (comparing to the one with Neumann condition at v).
2. A Neumann graph 􀀀 obeys the strong Dirichlet criterion with respect to its vertex v if it obeys the Dirichlet criterion and if imposing the Dirichlet vertex condition at v strictly increases the eigenvalue multiplicity of k1.
Theorem 3.2.6. Let G1; G2 be discrete graphs, let vi (i = 1; 2) be a vertex of Gi. Let G be the graph obtained by identifying v1 and v2. Let l(i) 2 LGi and 􀀀i := 􀀀(G; l(i)) be the corresponding metric graphs. Define l := (Ll(1); (1 􀀀 L) l(2)) 2 LG, for some L 2 [0; 1]. Then the graph 􀀀 := 􀀀(G; l) is a supremizer of G if all the following conditions are met.

Spectral gaps as critical values

In this section we assume that the spectral gap, k1 (􀀀 (G; l)), is a simple eigenvalue. This allows to take derivatives of the eigenvalue with respect to the edge lengths, l 2 LG, and to find critical points which serve as candidates for maximizers. We prove here Theorem 3.2.4 which shows that such local maximizers do not achieve a spectral gap higher than that achieved by turning the graph into a mandarin or a flower.
Lemma 3.5.1. Let 􀀀 be a metric graph and f an eigenfunction corresponding to the eigenvalue k2 with arbitrary vertex conditions. Then the function f0(x)2 + k2f(x)2 is constant along each edge. Démonstration. The proof is immediate, once differentiating the function f0(x)2+k2f(x)2 along an edge. The last lemma motivates us to define the energy 6 of an eigenfunction on an edge e as Ee := f0(x)2 + k2f(x)2 for any x 2 e. This energy shows up naturally when differentiating an eigenvalue with respect to an edge length. In order to evaluate such derivatives we extend Definition 3.1.1 so that 􀀀 (G; l) is defined for all l 2 RE with positive entries and relax the restriction PE e=1 le = 1, imposed by l 2 LG. The following lemma appears also as Lemma A.1 in [47] and within the proof of a lemma in [55].
Lemma 3.5.2. Let G be a discrete graph and let l 2 RE with positive entries. Assume that the spectral gap, k1 [􀀀(G; l)] is a simple eigenvalue and let f be the corresponding eigenfunction, normalized to have unit L2 norm. Then k1 [􀀀(G; l)] is differentiable with respect to any edge length l~e and @ @l~e (k1 [􀀀 (G; l)])2 = 􀀀E~e: (3.29).
Démonstration. In this proof we use the analyticity of the eigenvalues and eigenfunctions with respect to the edge lengths. This is established for example in sections 3.1.2, 3.1.3 of [43]. Let s 2 R and let ~e be an edge of 􀀀(G; l). Denote l (s) := l + s~e, with ~e 2 RE a vector with one at its ~eth position and zeros in all other entries. We use the notation 􀀀 (s) := 􀀀(G; l (s)) and denote by k1 (s) the spectral gap of 􀀀 (s). By assumption, k1 (0) is a simple eigenvalue and hence there is a neighborhood of zero for which all k1 (s) are simple eigenvalues. The corresponding eigenfunctions are denoted by f (s; ) and we further assume that all those eigenfunctions have unit L2 norm.

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Introduction générale
1 Mécanique des fluides incompressibles
1.1 Équations de Navier-Stokes-Euler homogènes incompressibles
1.2 Notions de solutions
1.3 Théorie de DiPerna-Lions
1.4 Résultats
2 Graphes quantiques
2.1 Bref historique
2.2 Définitions essentielles
2.3 Formulation du problème
2.4 Résultats
3 Groupes de Lie et transformée de Fourier
3.1 Groupes et algèbres de Lie
3.2 Groupes de Lie nilpotents
3.3 Représentations unitaires irréductibles
3.4 Transformation de Fourier
3.5 Décomposition en coefficients matriciels
3.6 Lien avec le laplacien
3.7 Résultats
I Mécanique des fluides incompressibles
1 Un lemme d’unicité et ses applications en mécanique des fluides incompressibles
1.1 Introduction
1.2 Results
1.3 Proofs
2 Sur un critère de Serrin anisotrope pour les solutions faibles des équations de Navier-Stokes
2.1 Presentation of the problem
2.2 Overview of the proof
2.3 Notations
2.4 Preliminary lemmas
2.5 Case of the torus
2.6 Local case in R3
II Graphes quantiques
3 Graphes quantiques optimisant leur trou spectral.
3.1 Introduction
3.1.1 Discrete graphs and graph topologies
3.1.2 Spectral theory of quantum graphs
3.1.3 Graph Optimizers
3.2 Main Results
3.2.1 Infimizers (section 3.3)
3.2.2 Supremizers of tree graphs (section 3.4)
3.2.3 Supremizers whose spectral gap is a simple eigenvalue (section 3.5)
3.2.4 Supremizers of vertex connectivity one (sections 3.6, 3.7, 3.8)
3.3 Infimizers
3.4 Supremizers of tree graphs
3.5 Spectral gaps as critical values
3.6 Gluing Graphs
3.7 Symmetrization of dangling edges and loops
3.8 Applications of graph gluing and symmetrization
3.9 Summary
III Transformée de Fourier
4 Transformée de Fourier sur les groupes de Lie nilpotents d’indice 2
4.1 Introduction
4.1.1 Definition of 2-steps Lie groups
4.1.2 A few examples
4.1.3 Definition of the Fourier transform
4.1.4 The frequency space
4.2 Description of the results
4.3 Topology and measure theory on ^g
4.3.1 The completion of the frequency space
4.3.2 A measure on ^g
4.4 A study of the Fourier kernel
4.4.1 Regularity and decay of
4.4.2 Continuous extension of W to ^g
4.5 The case of functions independant of the central variable
4.6 Computing the kernel at the boundary
4.6.1 Preliminary identities
4.6.2 Another expression for K
IV Annexes
A Graphes quantiques
A.1 Eigenvalue continuity with respect to edge lengths
A.1.1 The scattering approach to the graph spectrum
A.1.2 Continuity of eigenvalues via scattering approach
A.2 -type conditions and interlacing theorems
A.3 A basic Rayleigh quotient computation
A.4 Proofs for small stowers (Lemmata 3.8.1-3.8.5)
B Transformée de Fourier
B.1 Standard computations on the Hermite functions
B.2 The representation-theoretic Fourier transform. .

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