# Local bifurcation diagrams and degenerate solutions of Yamabe-type equations

Sahid Bernabé Catalán  Centro de Investigación en Matemáticas, CIMAT, Calle
Jalisco s/n, 36023 Guanajuato, Guanajuato, México<EMAIL_ADDRESS>and
Jimmy Petean  Centro de Investigación en Matemáticas, CIMAT, Calle Jalisco
s/n, 36023 Guanajuato, Guanajuato, México<EMAIL_ADDRESS>
###### Abstract.

We study positive solutions of the equation $-\Delta_{g}u+\lambda u=\lambda
u^{q}$, with $\lambda>0$, $q>1$ on the round sphere $\mathbb{S}^{n}$. We
reduce the equation to an ordinary differential equation by considering
isoparametric functions and apply bifurcation theory. We study when the
corresponding bifurcation points are transcritical. We apply this result to
show the existence of degenerate solutions to the equation and to study
multiplicity results for conformal constant scalar curvature metrics.

##  1\. Introduction

On a closed Riemannian manifold $(M^{n},g)$ of dimension $n\geq 3$ we consider
the following Yamabe type equation

(1.1) $-\Delta_{g}u+\lambda u=\lambda u^{q},$

with $\lambda>0$ and $q>1$. Let $p_{n}=\frac{n+2}{n-2}$. Equation (1.1) is
said to be critical if $q=p_{n}$, subcritical if $q<p_{n}$ and supercritical
if $q>p_{n}$. The equation is very well-known and has been extensively studied
in the last decades. The critical case appears in Riemannian geometry when
trying to solve the problem of finding metrics of constant scalar curvature in
a given conformal class of metrics; what is known as the Yamabe problem.
Namely, if a metric $h$ conformal to $g$ is expressed as $h=u^{p_{n}-1}g$ for
a positive function $u$ on $M$, then the scalar curvature of $h$, $s_{h}$, is
equal to a constant $\lambda\in{\bf R}$ if and only if $u$ solves the Yamabe
equation:

(1.2) $-a_{n}\Delta_{g}u+s_{g}u=\lambda u^{p_{n}},$

where $a_{n}=\frac{4(n-1)}{n-2}$. A fundamental result obtained in several
steps by H. Yamabe [30], N. Trudinger [28], T. Aubin [2] and R. Schoen [26]
states that there is always at least one positive solution of the Yamabe
equation, for any closed Riemannian manifold. Since the Yamabe equation is
conformally invariant, in order to understand the space of solutions we can
therefore assume that $s_{g}$ is constant. In the non-positive case Equation
(1.2) only has the constant solution and therefore we assume that $s_{g}$ is
positive. Then Equation (1.2) is of the form (1.1) with $q=p_{n}$ and
$\lambda=\frac{s_{g}}{a_{n}}$.

We will study Equation (1.1) on the sphere of dimension $n$ with the metric of
constant sectional curvature 1, $(\mathbb{S}^{n},g_{0})$. If we consider a
Riemannian product with the sphere, $(M\times\mathbb{S}^{n},g+g_{0})$, we
could search for solutions of the Yamabe equation which are functions that
depend only on the sphere. Similarly if a Riemannian manifold is the total
space of a harmonic Riemannian submersion over the sphere one can consider
solutions which are constant along the fibers. Then the Yamabe equation
reduces to a subcritical equation on the sphere (see for instance [4, 10,
24]). Therefore the results we will prove on positive solutions to Equation
(1.2) also give multiplicity results for conformal constant scalar curvature
metrics on the total spaces of such fibrations.

Many multiplicity results for the Yamabe type equation (1.1) have been
obtained using bifurcation techniques. In bifurcation theory one considers a
family of solutions (which one considers as the family of trivial solutions)
and tries to understand the connected component of this family of trivial
solutions (in the space of all solutions). In particular one wants to
understand which are the bifurcation points of the family; the elements of the
family which are accumulation points of solutions which are not in the family.
By Local Bifurcation we mean to understand which are the bifurcation points in
the family and what is the space of solutions in a neighborhood of a
bifurcation point. Local bifurcation for Equation (1.1) was studied for
instance in [4, 5, 10, 24, 25]. We mean by Global Bifurcation the study of the
whole connected component, which is a more difficult problem. Global
bifurcation results for Equation (1.1) have been obtained in [3, 6, 15]. All
these results on global bifurcation have been obtained using a reduction of
the equation to an ordinary dfferential equation, as we will do in this
article.

Recall that a function $f:M\rightarrow[t_{0},t_{1}]$, on a closed Riemannian
manifold $(M,g)$, is called isoparametric if $|\nabla f|^{2}=b(f)$, $\Delta
f=a(f)$, for some smooth functions $a,b:[t_{0},t_{1}]\rightarrow\mathbb{R}$.
Isoparametric functions on general Riemannian manifolds were considered by Q-M
Wang in [29], following the classical work by E. Cartan [8], B. Segre [27], T.
Levi-Civita [19] in the case of space forms. An isoparametric function is
called proper if its regular level sets are connected. For instance if by
$x_{1}:(\mathbb{S}^{n},g_{0})\rightarrow\mathbb{R}$ we mean the projection in
the first coordinate, then $x_{1}$ is a proper isoparametric function while
$x_{1}^{2}$ is an isoparametric fuction which is not proper. More details on
isoparametric functions can be found in [13, 14, 20, 21].

Assume that there is an isoparametric function $f:M\rightarrow\mathbb{R}$ and
look for solutions of Equation (1.1) of the form $v=\varphi\circ f$, where
$\varphi:\mathbb{R}\rightarrow\mathbb{R}_{>0}$. We will call functions $v$ of
this form $f$-invariant. We will denote by $C_{f}^{2,\alpha}$ the space of
$C^{2,\alpha}$ functions on $M$ which are $f$-invariant.

We will consider (1.1) as an equation on $(u,\lambda)$ and study solutions
bifurcating from the family of trivial solutions $\lambda\mapsto(1,\lambda)$.
In this context one says that the family of solutions is locally rigid at
$(1,\lambda_{0})$ if there is a neighborhood $U$ of $(1,\lambda_{0})$ such
that if $(u,\lambda)\in U$ is a solution then $u=1$. In the other case, namely
if there is a sequence of non-constant solutions approaching
$(1,\lambda_{0})$, we say that $(1,\lambda_{0})$ is a bifurcation point for
the family.

Using the well-known theory of local bifurcation for simple eigenvalues, which
started with the classical work of M. G. Crandall, P. H. Rabinowitz in [9],
one can prove the following result (it is proved explicitly in this case in
[3]):

######  Theorem 1.1.

Let $f$ be a proper isoparametric function on $(M,g)$. Let $\mu_{k}$, $k\geq
1$, be the (negative) eigenvalues of $\Delta_{g}|_{C_{f}^{2,\alpha}}$. For any
$q>1$ let $\lambda_{k}=\lambda_{k}(q)=\frac{-\mu_{k}}{q-1}$. For any $k\geq 1$
there is a continuous branch $t\mapsto(u(t),\lambda(t))$,
$t\in(-\varepsilon,\varepsilon)$, of $f$-invariant solutions of Equation (1.1)
so that $\lambda(0)=\lambda_{k}$, $u(0)=1$ and $u(t)\neq 1$ if $t\neq 0$.

We will consider the local behavior of the non-trivial branch of solutions
appearing at the bifurcation points. Recall that the bifurcation point
$(1,\lambda_{k})$ is called transcritical if the nontrivial branch
$s(t)=(u(t),\lambda(t))$ verifies $\lambda^{\prime}(0)\neq 0$. In this case
the branches $s(t)$, $t>0$ and $s(t)$, $t<0$, are on different sides of the
vertical line $\lambda(t)=\lambda_{k}$. If $\lambda^{\prime}(0)=0$ and
$\lambda^{\prime\prime}(0)\neq 0$ then both branches stay on the same region
$\lambda<\lambda_{k}$ or $\lambda>\lambda_{k}$.

From now on we will focus on the case of the round sphere
$(\mathbb{S}^{n},g_{0})$. The simplest isoparametric functions on the sphere
are obtained by considering the isometric cohomogeneity one $O(n)$-action
fixing an axis. The $O(n)$-invariant functions will be called radial (with
respect to the fixed axis). A linear function on $\mathbb{R}^{n+1}$ invariant
by the action (and restricted to the sphere) gives a proper isoparametric
function on the sphere. It is actually an eigenfunction for the Laplacian,
corresponding to the first nontrivial eigenvalue. The first result about the
local behavior of the non-trivial branches of solutions in this situation was
obtained by H. Brezis and Y.Y. Li in [7]. They considered the case of the
first radial eigenfunction, $\lambda_{1}$. It was shown in [7, Theorem 3,
Remarks 7 and 8] that if $(u(s),\lambda(s))$ is the branch of non-trivial
solutions bifurcating at $(1,\lambda_{1})$, then $\lambda^{\prime}(0)=0$.
Moreover $\lambda^{\prime\prime}(0)\neq 0$, and the sign of
$\lambda^{\prime\prime}(0)$ depends on whether the equation is sub or
supercritical.

General isoparametric functions on the sphere are the restrictions of Cartan-
Münzner polynomials on $\mathbb{R}^{n+1}$ ([20, 21]). For a polynomial $F$ in
$\mathbb{R}^{n+1}$ we let $d$ be the degree of $F$. Then $F$ is called a
Cartan-Münzner polynomial if it satisfies the Cartan-Münzner equations:

$\langle\nabla F,\nabla F\rangle=d^{2}\|x\|^{2d-2}$

$\Delta F=\frac{1}{2}cd^{2}\|x\|^{d-2},$

where $c$ is a constant, which we will describe below. Then $f=F_{|_{S^{n}}}$
is an isoparametric function on the sphere: it verifies

$\langle\nabla f,\nabla f\rangle=d^{2}(1-f^{2})$

$\Delta f=-d(n+d-1)f+\frac{1}{2}cd^{2}.$

The constant $c$ has a geometric meaning (see [20, 21]):
$f:S^{n}\rightarrow[-1,1]$ and its only critical values are -1 and 1. For
$t\in(-1,1)$ $f^{-1}(t)$ is called an isoparametric hypersurface of degree
$d$. $d$ can only take the values 1, 2, 3, 4 or 6. $f^{-1}(t)$ has constant
principal curvatures, and the number of distinct principal curvatures is $d$.
In case $d$ is odd all the distinct principal curvatures have the same
multiplicity. If $d=2,4$ or 6 then half of principal curvatures have
multiplicity $m_{1}$ and the other half have multiplicity $m_{2}$. The
constant $c$ is $m_{2}-m_{1}$. Note that $-F$ also satisfies the Cartan-
Münzner equations replacing $c$ by $-c$, which amounts to exchanging $m_{1}$
and $m_{2}$. Then we will always assume that $c\leq 0$.

We will denote an isoparametric function on the sphere by $f^{d,c}$ (even when
there are different Cartan-Münzner polynomials with the same values of $d$ and
$c$).

Note that if $\varphi\in C^{2}[-1,1]$ then $\Delta(\varphi\circ
f^{d,c})=d^{2}(1-(f^{d,c})^{2})\ (\varphi^{\prime\prime}\circ f^{d,c})\
+(-d(n+d-1)f^{d,c}+(1/2)cd^{2})\ (\varphi^{\prime}\circ f^{d,c})$. And
therefore $\varphi\circ f^{d,c}$ satisfies Equation (1.1) if and only if
$\varphi$ satisfies

(1.3)
$d^{2}(1-t^{2})\varphi^{\prime\prime}(t)+(-d(n+d-1)t+(1/2)cd^{2})\varphi^{\prime}(t)-\lambda(\varphi(t)-\varphi(t)^{q})=0.$

Recall that the eigenvalues of the Laplace operator on the sphere are
$\mu_{i}=-i(n+i-1)$. The eigenvalues of the Laplace operator restricted to the
space of $f^{d,c}$-invariant functions are $\mu_{di}=-di(n+di-1)$ (see [15,
Lemma 3.4]). The space of $f^{d,c}$-invariant eigenfunctions corresponding to
the eigenvalue $\mu_{di}$ has dimension 1: let $\varphi_{i}(d,c)\neq 0$ be
such an eigenfunction and write $\varphi_{i}(d,c)=\varphi\circ f^{d,c}$. Then
$\varphi$ satisfies

(1.4)
$d^{2}(1-t^{2})\varphi^{\prime\prime}(t)+(-d(n+d-1)t+(1/2)cd^{2})\varphi^{\prime}(t)+di(n+di-1)\varphi(t)=0.$

If we pick $\alpha,\beta$ by solving

$\beta-\alpha=(1/2)c$

$\alpha+\beta+2=\frac{n+d-1}{d},$

Then Equation (1.4) becomes

(1.5)
$(1-t^{2})\varphi^{\prime\prime}(t)+(-(\alpha+\beta+2)t+\beta-\alpha)\varphi^{\prime}(t)+i(i+\alpha+\beta+1)\varphi(t)=0,$

which is the usual form of the classical Jacobi equation. Note that since we
are assuming $c\leq 0$ we have that $\alpha\geq\beta$ and we also have that
$\alpha+\beta+1>0$.

Let $(u^{d,c}_{i}(s),\lambda^{d,c}_{i}(s))$ be the branch of non-trivial
$f^{d,c}$-invariant solutions bifurcating at $(1,\frac{-\mu_{di}}{q-1})$, as
in Theorem 1.1. It is shown in [25, Section 3] that

$\frac{d\lambda^{d,c}_{i}}{ds}(0)=C\int_{\mathbb{S}^{n}}\varphi^{3},$

where $C\neq 0$ is a constant and $\varphi\neq 0$ is a $f^{k,c}$-invariant
eigenfunction of the Laplacian, $-\Delta\varphi=ki(n+ki-1)\varphi$. Using
this, it is shown in [25] that for the first associated eigenfunction one has
$\frac{d\lambda_{1}^{d,c}}{ds}(0)=0$ if and only if $c=0$. And in case $c=0$,
one has that, for the second associated eigenfunction,
$\frac{d\lambda^{d,0}_{2}}{ds}(0)\neq 0$.

We will prove:

######  Theorem 1.2.

If $c=0$ and $i\geq 1$ the branch of nontrivial bifurcating solutions
$(u^{d,0}_{i}(s),\lambda^{d,0}_{i}(s))$ of Equation (1.1) appearing at
$(1,\frac{-\mu_{di}}{q-1})$ verifies $\frac{d\lambda^{d,0}_{i}}{ds}(0)=0$ if
and only if $i$ is odd.

If $c\neq 0$ then for any $i$ the branch of nontrivial solutions
$(u^{d,c}_{i}(s),\lambda^{d,c}_{i}(s))$ appearing at
$(1,\frac{-\mu_{di}}{q-1})$ verifies $\frac{d\lambda^{d,c}_{i}}{ds}(0)\neq 0$.

The proof of Theorem 1.2 is very different from the one used in [25] for the
cases mentioned above. Note that

$\varphi_{1}(d,c)=f^{d,c}-\frac{cd^{2}}{2(n+d-1)},$

is a $\mu_{d}$-eigenvalue on the sphere, and also an isoparametric function.
Using this to compute $\Delta\varphi_{1}(d,c)^{2}$ and integration by parts it
is shown in [25] that

$(\mu_{d}+2d^{2})\int_{\mathbb{S}^{n}}\varphi_{1}(d,c)^{3}=-\frac{2cd^{3}}{n+d-1}\int_{\mathbb{S}^{n}}\varphi_{1}(d,c)^{2},$

which implies the first above mentioned result. When $c=0$ then one can see
that $\varphi_{2}(d,0)$ is also an isoparametric function (although it is not
proper). For instance in the case of the $O(n)$-invariant isoparametric
functions discussed above, we have that $\varphi_{1}(d,0)$ is a linear
function, $x$, and $\varphi_{2}(d,0)$ is $x^{2}-1/(n+1)$. Then in a similar
way it is shown in [25] that

$\int_{\mathbb{S}^{n}}\varphi_{2}(d,0)^{3}\neq 0.$

It is easy to see that all the other $f^{k,c}$-invariant eigenfunctions are
not isoparametric, and therefore the previous argument does not work. Instead,
in the article we will work directly with Equation (1.3), and we will prove
Theorem 1.2 by computing certain integrals related to the Jacobi polynomials.
This will be carried out in Section 3. In Section 2 we will have a discussion
on the Jacobi equations and polynomials, and their linearization formulas,
that are necessary to compute the required integrals.

In Section 4 we will discuss how to apply Theorem 1.2 to prove existence of
degenerate solutions of Equation (1.1). Recall that a solution is called
degenerate if the linearized equation has a nontrivial kernel. For instance,
in the context of Theorem 1.1 a constant solution $(1,\lambda)$ is degenerate
if and only if $(1,\lambda)$ is a bifurcation point. Degenerate solutions are
important to understand the global picture of the family of solution which
appear through bifurcation, since the turning points of the curves of
nontrivial solutions are degenerate solutions. We will need to impose
conditions on $q$. For any isoparametric function $f^{d,c}$ the set of
critical points has two connected components $M_{1}=(f^{d,c})^{-1}(-1)$ and
$M_{2}=f(^{d,c})^{-1}(1)$. Let $m_{i}$ is the dimension of $M_{i}$ and let
$m=\min\\{m_{1},m_{2}\\}\leq n-2$. Then we let $q_{f}=\frac{n-m+2}{n-m-2}$,
$q_{f}=\infty$ in case $m=n-2$. Note that if $m>0$ then $q_{f}>p_{n}$. For the
next result we will ask that $q<q_{f}$. If $m>0$ the result applies then to
supercritical equations.

We will prove:

######  Theorem 1.3.

Let $f^{d,c}$ be an isoparametric function on the sphere as above. If $c=0$,
then for any positive even integer $k$ there exists a degenerate
$f^{d,0}$-invariant positive solution of Equation (1.1) (for some $\lambda>0$)
for which the set of critical points has exactly $k+1$ connected components.

If $c\neq 0$ then for any positive integer $k$ there exists a degenerate
$f^{d,c}$-invariant solution of Equation (1.1) (for some $\lambda>0$) for
which the set of critical points has exactly $k+1$ connected components.

######  Remark 1.4.

A $f^{d,c}$-invariant positive solution, $\varphi\circ f^{d,c}$, of Equation
(1.1) is given by a positve solution, $\varphi$, of Equation (1.3). The
critical points of $\varphi\circ f^{d,c}$ are the critical points of
$f^{d,c}$, which has the two connected components $M_{1}$ and $M_{2}$
mentioned above, and the preimage by $f^{d,c}$ of the critical points of
$\varphi$. Therefore Theorem 1.3 is equivalent to prove the existence of
positive solutions of Equation (1.3) with $k-1$ critical points. Note that for
non-constant positive solutions $\varphi$ of Equation (1.3), 1 and -1 are not
critical ponts.

######  Remark 1.5.

Once we fixed a solution of the linearized Equation (1.4) we will actually
compute the sign of $\frac{d\lambda^{d,c}_{i}}{ds}(0)$. In this way we will
actually be able to say if the corresponding degenerate solution from Theorem
1.3 has a local maximum or a local minimum at $M_{1}$ and $M_{2}$.

##  2\. Jacobi polynomials and their linearization coefficients

In this section we will review results about Jacobi polynomials. This will
allow us to give the computations of certain integrals involving the Jacobi
polynomials, which are necessary to understand the local behaviour of the
bifurcation branches discussed in the introduction and in the next section.
Most of the results we will mention appear in the book by R. Askey [1], and we
will follow the same notation and normalizations used in the book.

The Jacobi differential equation is

(2.1)
$(1-t^{2})y^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)y^{\prime}(t)+k(k+\alpha+\beta+1)y(t)=0.$

In the equation $k$ is a positive integer, and $\alpha,\beta>-1$ are real
numbers. The equation is defined in the interval $[-1,1]$. Solutions are
$C^{2}$-functions which verify

(2.2)
$(\beta-\alpha+(\alpha+\beta+2))y^{\prime}(-1)+k(k+\alpha+\beta+1)y(-1)=0$

and

(2.3)
$(\beta-\alpha-(\alpha+\beta+2))y^{\prime}(1)+k(k+\alpha+\beta+1)y(1)=0.$

It is of course an eigenvalue problem. For $\lambda\in\mathbb{R}$ consider the
equation

(2.4)
$(1-t^{2})y^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)y^{\prime}(t)+\lambda
y(t)=0.$

The space of solutions of Equation (2.4) satisfying the initial condition
(2.2) has dimension 1. One looks for the values of $\lambda$ for which the
corresponding solutions are defined in the whole interval [-1,1], and satisfy
(2.3). Constant functions satisfy Equation (2.4) for $\lambda=0$. It is also
easy to check that (multiples of) $y(t)=t-\frac{\beta-\alpha}{\alpha+\beta+2}$
satisfies Equation (2.4) for $\lambda=1(1+\alpha+\beta+1)$.

Let
$L(y)=(1-t^{2})y^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)y^{\prime}(t)$.
Then we point out that for $k\geq 2$ we have that

$L(t^{k})=-k(k+\alpha+\beta+1)t^{k}+k(\beta-\alpha)t^{k-1}+k(k-1)t^{k-2}.$

Note that $L(t^{k})+k(k+\alpha+\beta+1)t^{k}$ is a polynomial of degree $k-1$
and if $k\neq j$ then $L(t^{k})+j(j+\alpha+\beta+1)t^{k}$ is a polynomial of
degree $k$.

It then follows easily by induction that for each $k$ there is a polynomial of
degree $k$ satisfying Equation (2.1). Such polynomial is called a Jacobi
polynomial, denoted as $P_{k}^{(\alpha,\beta)}$.

######  Remark 2.1.

Note that with the same argument one can see that if $\alpha=\beta$ then if
$k$ is odd then $P_{k}^{(\alpha,\beta)}$ is an odd polynomial and if $k$ is
even then $P_{k}^{(\alpha,\beta)}$ is an even polynomial. This is not the case
if $\alpha\neq\beta$, as we have seen for instance for
$P_{1}^{(\alpha,\beta)}$.

For $\alpha,\beta$ fixed, the Jacobi polynomials $P_{k}^{(\alpha,\beta)}$,
$k\geq 1$, are orthogonal with respect to the weight
$(1-t)^{\alpha}(1+t)^{\beta}$: i.e. if $j\neq k$ then

$\int_{-1}^{1}P_{k}^{(\alpha,\beta)}P_{j}^{(\alpha,\beta)}(1-t)^{\alpha}(1+t)^{\beta}dt=0.$

One can check this easily. First note that for any $u,v\in C^{2}[-1,1]$
integrating by parts one obtains that:

(2.5) $\int_{-1}^{1}((1-t)^{\alpha+1}(1+t)^{\beta+1}u^{\prime})^{\prime}\ \ v\
\ dt=\int_{-1}^{1}((1-t)^{\alpha+1}(1+t)^{\beta+1}v^{\prime})^{\prime}\ \ u\ \
dt.$

Also note that:

$((1-t)^{\alpha+1}(1+t)^{\beta+1}u^{\prime})^{\prime}=(1-t)^{\alpha}(1+t)^{\beta}\left[(1-t^{2})u^{\prime\prime}-(\alpha+1)(1+t)u^{\prime}+(\beta+1)(1-t)u^{\prime}\right]$

$=(1-t)^{\alpha}(1+t)^{\beta}((1-t^{2})u^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)t)u^{\prime}).$

In particular for $u=P_{k}^{(\alpha,\beta)}$ we get

$((1-t)^{\alpha+1}(1+t)^{\beta+1}(P_{k}^{(\alpha,\beta)})^{\prime})^{\prime}=-k(k+\alpha+\beta+1)(1-t)^{\alpha}(1+t)^{\beta}P_{k}^{(\alpha,\beta)},$

and setting $u=P_{k}^{(\alpha,\beta)}$, $v=P_{j}^{(\alpha,\beta)}$ in (2.5),
with $j\neq k$, we obtain

$k(k+\alpha+\beta+1)\int_{-1}^{1}P_{k}^{(\alpha,\beta)}P_{j}^{(\alpha,\beta)}(1-t)^{\alpha}(1+t)^{\beta}dt=j(j+\alpha+\beta+1)\int_{-1}^{1}P_{j}^{(\alpha,\beta)}P_{k}^{(\alpha,\beta)}(1-t)^{\alpha}(1+t)^{\beta}dt=0.$

Since $P_{k}^{(\alpha,\beta)}$ has degree $k$ it follows that
$(P_{k}^{(\alpha,\beta)})_{k\geq 0}$ forms a complete orthogonal system for
the weighted $L^{2}$-space and therefore $k(k+\alpha+\beta+1)$, $k\geq 0$ are
all the eigenvalues for the eigenvalue problem (2.4).

Gegenbauer polynomials are particular cases of Jacobi polynomials, obtained
when $\alpha=\beta$. For $\alpha>-1/2$ we call
$C_{k}^{\alpha}=P_{k}^{\alpha-(1/2),\alpha-(1/2)}$. The Gegenbauer polynomial
$C_{k}^{\alpha}$ has degree $k$ and satisfies the Gegenbauer differential
equation

(2.6)
$(1-t^{2})y^{\prime\prime}(t)-(2\alpha+1)t)y^{\prime}(t)+k(k+2\alpha)y(t)=0.$

The Legendre polynomials $P_{k}$ are particular cases of Gegenbauer
polynomials, taking $\alpha=1/2$: $P_{k}=C_{k}^{1/2}=P_{k}^{0,0}$.

All Jacobi polynomials can be computed recursively as it was discussed above.
One also has the explicit formula [1, Lecture 2]

$P_{k}^{(\alpha,\beta)}(t)=(1-t)^{-\alpha}(1+t)^{-\beta}\frac{(-1)^{k}}{2^{k}k!}\frac{d^{k}}{dt^{k}}((1-t)^{k+\alpha}(1+t)^{k+\beta})$

We will use this formula to fix a normalization for $P_{k}^{(\alpha,\beta)}$
as in [1]. Note that with this normalization we have that
$P_{0}^{(\alpha,\beta)}(t)=1$,
$P_{1}^{(\alpha,\beta)}(t)=(1/2)(\alpha-\beta)+(t/2)(\alpha+\beta+2)$. Also:

$P_{k}^{(\alpha,\beta)}(-1)=(-1)^{k}\frac{(\beta+1)_{k}}{k!}=(-1)^{k}\frac{(\beta+1)...(\beta+k)}{k!},$

and

$P_{k}^{(\alpha,\beta)}(1)=\frac{(\alpha+1)_{k}}{k!}=\frac{(\alpha+1)...(\alpha+k)}{k!}.$

In particular $P_{k}^{(\alpha,\beta)}(1)>0$ while
$P_{k}^{(\alpha,\beta)}(-1).P_{k+1}^{(\alpha,\beta)}(-1)<0$. By a simple
application of Sturm-Liouville comparison theorem, the number of zeroes in
(-1,1) of a nontrivial solution of Equation (2.4) is nondecreasing in
$\lambda$. Since $P_{1}^{(\alpha,\beta)}$ has a zero in
$\frac{\beta-\alpha}{\alpha+\beta+2}\in(-1,1)$, it follows by induction that
$P_{k}^{(\alpha,\beta)}$ has $k$ zeroes in $(-1,1)$.

The linearization problem for the family $P_{k}^{(\alpha,\beta)}$, $k\geq 0$,
of Jacobi polynomials consists in finding the coefficients $C_{jk}^{l}$ such
that

$P_{k}^{(\alpha,\beta)}P_{j}^{(\alpha,\beta)}=\sum_{l=0}^{j+k}C_{jk}^{l}P_{l}^{(\alpha,\beta)}$

The linearization coefficients can be obtained for any particular case, but it
is difficult to obtain general formulas. There are expressions for the
linearization coefficients using hypergeometric functions, but it is difficult
to obtain general statements from these. We will summarize now a few things
that are known. For more details see the discussion in [1, Lecture 5].

The most classical case of Legendre polynomials is well understood. One of the
methods used to obtain the linearization coefficients is to obtain the fourth
order differential equation satisfied by a product $P_{k}P_{j}$. Then plug in
$\sum_{l=0}^{j+k}C_{jk}^{l}P_{l}^{(\alpha,\beta)}$ into the equation and use
known relationships between the polyomials to obtain the linearization
coefficients. This is done in the classical book by E. W. Hobson [16, Capter
2]. The argument goes back to the work of F. E. Neumann [22].

The same argument applies for the linearization coefficients for the
Gegenbauer polynomials. The formula for the linearization coefficients in this
case is known as Dougall´s formula. It was stated without a proof by J.
Dougall in [11]. A proof of the formula was given by H. Y. Hsü in [17]. E.
Hyllerass [18] gave a proof of the formula using an argument similar to the
one mentioned above for the case of the Legendre polynomials. In Dougall’s
formula the linearizarion coefficients appear as simple products and so it is
easy to check the sign of the coefficients. Such simple formula is not known
for general Jacobi polynomials. But Hylleraas obtained the differential
equation satisfied by the product of general Jacobi polynomials and deduced
relationships between the coefficients which help to understand them.

Later G. Gasper [12] used these relationships to prove that if
$\alpha\geq\beta$ and $\alpha+\beta+1\geq 0$ then all the linearization
coefficients are nonnegative. It is easy to check using Gasper arguments when
the coefficients are actually strictly positive. To make it simpler we will
only consider the case of the square ${P_{k}^{(\alpha,\beta)}}^{2}$, since it
is the case that we will need.

In the next theorem we will discuss how to prove this using Gasper’s work, and
we also state all the results mentioned before and that we will need in the
following section.

######  Theorem 2.2.

Fix $\alpha,\beta>-1$ such that $\alpha\geq\beta$ and $\alpha+\beta+1>0$.
Consider the family of Jacobi polynomials $P_{k}^{(\alpha,\beta)}$, $k\geq 0$.
$P_{k}^{(\alpha,\beta)}$ has $k$ zeros in $(-1,1)$.
$P_{k}^{(\alpha,\beta)}(1)>0$ while $P_{k}^{(\alpha,\beta)}(-1)>0$ if $k$ is
even and $P_{k}^{(\alpha,\beta)}(-1)<0$ if $k$ is odd.

Let
$P_{k}^{(\alpha,\beta)}P_{k}^{(\alpha,\beta)}=\sum_{i=0}^{2k}C_{k}^{i}P_{i}^{(\alpha,\beta)}$.

A. If $\alpha=\beta$ then: if $k$ is odd then $C_{k}^{i}=0$ if $i$ is odd. If
$k$ is even then $C_{k}^{i}>0$ if $i$ is even. In particular since the family
is orthogonal with respect to the weight $(1-t)^{\alpha}(1+t)^{\beta}$ we
have:

If $k$ is odd

$\int_{-1}^{1}{P_{k}^{(\alpha,\beta)}}^{3}(1-t)^{\alpha}(1+t)^{\beta}\ dt=0.$

If $k$ is even

$\int_{-1}^{1}{P_{k}^{(\alpha,\beta)}}^{3}(1-t)^{\alpha}(1+t)^{\beta}\ dt>0.$

B. If $\alpha>\beta$ then $C_{k}^{i}>0$ for all $i=0,...,2k$. In particular

$\int_{-1}^{1}{P_{k}^{(\alpha,\beta)}}^{3}(1-t)^{\alpha}(1+t)^{\beta}\ dt>0.$

###### Proof.

The first statements have already been discussed in this section. Also for the
statement (A) we have pointed out that when $\alpha=\beta$, which is the case
of the Gegenbauer polynomials, $P_{k}^{(\alpha,\beta)}$ is odd if $k$ is ood
and even if $k$ is even. Therefore if $k$ is odd,
$[P_{k}^{(\alpha,\beta)}]^{2}$ is an even polynomial and therefore if is
written as a linear combination of $P_{2i}^{(\alpha,\beta)}$, $i\geq 0$. This
means $C_{k}^{i}=0$ if $i$ is odd. Then

$\int_{-1}^{1}{P_{k}^{(\alpha,\beta)}}^{3}(1-t)^{\alpha}(1+t)^{\beta}dt=\int_{-1}^{1}{P_{k}^{(\alpha,\beta)}}\left(\sum_{i=0}^{2k}C_{k}^{i}P_{i}^{(\alpha,\beta)}\right)(1-t)^{\alpha}(1+t)^{\beta}dt$

$=C_{k}^{k}\int_{-1}^{1}{P_{k}^{(\alpha,\beta)}}^{2}(1-t)^{\alpha}(1+t)^{\beta}dt=0.$

For the case when $\alpha=\beta$ and $k$ is even and the case when
$\alpha>\beta$ we will follow the argument in [12]. Let

$R_{k}^{(\alpha,\beta)}=\frac{P_{k}^{(\alpha,\beta)}}{P_{k}^{(\alpha,\beta)}(1)}.$

$R_{k}^{(\alpha,\beta)}$ is of course just a different normalization for the
Jacobi polynomial; note that it is a positive multiple of
$P_{k}^{(\alpha,\beta)}$ and therefore the sign of the linearization
coefficients for $R_{k}^{(\alpha,\beta)}$ are the same as the sign of the
corresponding linearization coefficients for $P_{k}^{(\alpha,\beta)}$. To
simplify the notation we will consider $k$ fixed and write

$R_{k}^{(\alpha,\beta)}R_{k}^{(\alpha,\beta)}=\sum_{i=0}^{2k}G^{i}R_{i}^{(\alpha,\beta)}.$

It is pointed out in [12, Page 173] that $G^{i}$ is a positive multiple of
$d_{i}=(-1)^{i}c_{i}$, where $c_{i}$ are the linearization coefficients of
hypergeometric functions treated in [18]. Using the recurrence formula given
by [18, (4.13)], G. Gasper obtained the following recurrence relation for the
coefficients $d_{i}$ [12, formula (5)]: let $a=\alpha+\beta+1>0$ and
$b=\alpha-\beta\geq 0$. For $1\leq j\leq 2k-1$ we have

$\frac{(j+1)(2j+1+a+b)(2k+j+1+a)(2k-j-1+a)(j+1)}{(2j+1+a)(2j+2+a)}d_{j+1}$

$=b\left(\frac{(j+1)^{2}(2k+j+2a)(2k-j)}{2j+1+a}-\frac{j^{2}(2k+j-1+2a)(2k-j+1)}{2j-1+a}\right)\
d_{j}$

$+\frac{(j-1+a)(2j+2\beta)(2k+j-1+2a)(2k-j+1)(j-1+a)}{(2j-2+a)(2j-1+a)}\
d_{j-1}.$

It is also proved by G. Gasper [12, page 174] that $d_{0}>0$ and $d_{2k}>0$.
Let $A_{j}$, $B_{j}$, $C_{j}$, be the coefficients multiplying $d_{j+1},d_{j}$
and $d_{j-1}$ in the previous formula, respectively. Note that $A_{j}>0$ and
$C_{j}>0$.

In the case of Gegenbauer polynomials, $b=0$, and it follows that $d_{i}>0$ if
$i$ is even, while $d_{i}=0$ if $i$ is odd. This completes the proof of
statement (A).

Assume now that $b>0$.

For the cases $j=2k$ we have

$b\frac{4k^{2}(4k-1+2a)}{4k-1+a}\
d_{2k}=\frac{(2k-1+a)(4k+2\beta)(4k-1+2a)(2k-1+a)}{(4k-2+a)(4k-1+a)}\
d_{2k-1}.$

And for the case $j=0$ we have

$\frac{(1+a+b)(2k+1+a)(2k-1+a)}{(1+a)(2+a)}d_{1}=b\frac{(2k+2a)(2k)}{1+a}\
d_{0}.$

It follows directly from these that $d_{1}>0$ and $d_{2k-1}>0$. In particular
this already implies statement (B) for the case $k=1$. Assume then that $k>1$.

To study the sign of $B_{j}$ write

$B_{j}=\frac{Q_{j}}{(2j+1+a)(2j-1+a)}.$

We need to understand the sign of $Q_{j}$. It will be useful to let $J=j-1$.
We have by a simple computation that $Q_{J}$ is a polynomial of degree 4 in
$J$:

$Q_{J}=(J+2)^{2}(J+2k+2a+1)(2k-J-1)(2J+a+1)-(J+1)^{2}(J+2k+2a)(2k-J)(2J+a+3)$

$=-6J^{4}-12(a+2)J^{3}+(8k(k+a)-6a^{2}-38a-34)J^{2}$

$+(8k(k+a)(a+2)-14a^{2}-38a-20)J+4(k+3ka+3a+1)(k-1)+4ak+4a^{2}(3k-2).$

The coefficients multiplying $J^{4}$ and $J^{3}$ are negative. Since we
assumed that $k\geq 2$, we have that the coefficients multiplying $J^{1}$ and
$J^{0}$ are positive. Independently of the sign of the coefficient multiplying
$J^{2}$ we have that the coefficients change sign exactly once, going from
positive no negative either at $J^{2}$ or at $J^{3}$. Then it follows from
Descartes’ rule of signs that $Q_{J}$, as a function of a real variable, has
exactly one positive zero. since $Q(0)>0$ and $Q(+\infty)=-\infty$, it follows
that there exists $x_{0}>0$ sucht that $Q(x)>0$ if $x\in(0,x_{0})$ and
$Q(x)<0$ if $x>x_{0}$.

Note that if $B_{j}\geq 0$ then it follows that if $d_{j-1},d_{j}>0$ then
$d_{j+1}>0$. If $B_{j}<0$ then it follows that if $d_{j+1},d_{j}>0$ then
$d_{j-1}>0$. Then if $B_{j}>0$ for all $1\leq j\leq 2k-1$ it follows that
$d_{j}>0$ for all $0\leq j\leq 2k$. And also if $B_{j}<0$ for all $1\leq j\leq
2k-1$ then $d_{j}>0$ for all $0\leq j\leq 2k$. More generally, if
$B_{j_{0}}\geq 0$ for some $1\leq j_{0}\leq 2k-1$ (which implies form the
previous discussion that $B_{j}>0$ for all $j<j_{0}$) then $d_{j}>0$ for all
$1\leq j\leq j_{0}+1$. And if $B_{j_{0}+1}<0$ for some $1\leq j_{0}\leq 2k-2$
(which implies from the previous discussion that $B_{j}<0$ for all $j\geq
j_{0}+1$) then $d_{j}>0$ for all $j_{0}\leq j\leq 2k-1$. The previous
discussion says that there is a $j_{0}$ verifying these two conditions, and
therefore again $d_{j}>0$ for all $j$. This completes the proof of statement
(B).

∎

##  3\. Local bifurcation diagrams

We will consider an equation of the form

(3.1)
$(1-t^{2})u^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)u^{\prime}(t)-\lambda(u-u^{q})(t)=0.$

where $\alpha,\beta>-1$, $\lambda>0$, $q>1$. The equation is considered on the
interval $[-1,1]$ and the solution $u$ must satisfy the boundary conditions

(3.2) $(2\beta+2)u^{\prime}(-1)-\lambda(u(-1)-u(-1)^{q})=0.$

(3.3) $-(2\alpha+2)u^{\prime}(1)-\lambda(u(1)-u(1)^{q})=0.$

We consider the space $X=C^{2,\alpha}[-1,1]\times(0,\infty)$. And
$Y=C^{0,\alpha}[-1,1]$. Then we consider the map $F:X\rightarrow Y$,
$F(u,\lambda)=(1-t^{2})u^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)u^{\prime}(t)-\lambda(u-u^{q})(t)$.

We will consider positive functions $u$, actually close to the constant
solution $u=1$ (solution of $F(u,\lambda)=0$). Note that, for $v\in
C^{2}[-1,1]$,

$D_{u}F(u,\lambda)[v]=(1-t^{2})v^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)v^{\prime}(t)-\lambda(1-qu^{q-1})v(t).$

In particular

$D_{u}F(1,\lambda)[v]=(1-t^{2})v^{\prime\prime}(t)+(\beta-\alpha-(\alpha+\beta+2)t)v^{\prime}(t)-\lambda(1-q)v(t).$

######  Lemma 3.1.

For any $v,w\in C^{2}[-1,1]$

(3.4) $\int_{-1}^{1}D_{u}F(1,\lambda)[v]\ w(1-t)^{\alpha}(1+t)^{\beta}\
dt=\int_{-1}^{1}D_{u}F(1,\lambda)[w]\ v(1-t)^{\alpha}(1+t)^{\beta}\ dt$

###### Proof.

By a direct computation

$((1-t)^{\alpha+1}(1+t)^{\beta+1}v^{\prime})^{\prime}=(1-t)^{\alpha}(1+t)^{\beta}((1-t^{2})v^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)t)\
v^{\prime}),$

and therefore

$\int_{-1}^{1}D_{u}F(1,\lambda)[v]\ w(1-t)^{\alpha}(1+t)^{\beta}\
dt=\int_{-1}^{1}((1-t)^{\alpha+1}(1+t)^{\beta+1}v^{\prime})^{\prime}w-\lambda(1-q)vw(1-t)^{\alpha}(1+t)^{\beta}\
dt$

We apply integration by parts as in (2.5) to the first term on the right to
obtain:

$\int_{-1}^{1}D_{u}F(1,\lambda)[v]\ w(1-t)^{\alpha}(1+t)^{\beta}\
dt=\int_{-1}^{1}((1-t)^{\alpha+1}(1+t)^{\beta+1}w^{\prime})^{\prime}v-\lambda(1-q)vw(1-t)^{\alpha}(1+t)^{\beta}\
dt$

$=\int_{-1}^{1}D_{u}F(1,\lambda)[w]\ v(1-t)^{\alpha}(1+t)^{\beta}\ dt.$

∎

The following results gives a more explicit discussion for the local
bifurcation for Equation (3.1) and contains Theorem 1.2.

######  Theorem 3.2.

Let $\lambda_{k}=\frac{k(k+\alpha+\beta+1)}{q-1}$. If $\lambda\neq\lambda_{k}$
for all $k$ then the trivial family of solutions of Equation (3.1) is locally
rigid at $(1,\lambda)$. For each $k\geq 1$, $(1,\lambda_{k})$ is a bifurcation
point for the family. The set of nontrivial solutions in a neighborhood of
$(1,\lambda_{k})$ is given by a path $s\mapsto(u(s),\lambda(s))$, such that
$u(0)=1$, $\lambda(0)=\lambda_{k}$, $u(s)\neq 1$ if $s\neq 0$, and
$\frac{du}{ds}(0)=P_{k}^{(\alpha,\beta)}$ (the Jacobi polynomial from the
previous section). We have:

$\frac{d\lambda}{ds}(0)=C\int_{-1}^{1}(P_{k}^{(\alpha,\beta)})^{3}(1-t)^{\alpha}(1+t)^{\beta}\
dt,$

where

$C=\frac{-q}{2\int_{-1}^{1}(P_{i}^{\alpha,\beta})^{2}(1-t)^{\alpha}(1+t)^{\beta}dt}$

is a negative constant. Therefore if $\alpha>\beta$ or $\alpha=\beta$ and $k$
is even we have that $\frac{d\lambda}{ds}(0)<0$. If $\alpha=\beta$ and $k$ is
odd then $\frac{d\lambda}{ds}(0)=0$.

###### Proof.

We apply the classical resut on bifurcation from simple eigenvalues (see [9,
23]) to the operator $F:X\rightarrow Y$. $ND_{u}F(1,\lambda_{k})=\langle
P_{k}^{\alpha,\beta}\rangle$ and it follows from Lemma 3.1 that
$D_{u}F(1,\lambda_{k})$ is self adjoint with respect to the
$(1-t)^{\alpha}(1+t)^{\beta}$-weighted $L^{2}$ product on $[-1,1]$. The range
of $D_{u}F(1,\lambda_{k})$, $R$, is the space orthogonal to
$P_{k}^{\alpha,\beta}$ and
$D_{u\lambda}F(1,\lambda_{k})[P_{k}^{\alpha,\beta}]=-(1-q)P_{i}^{\alpha,\beta}\notin
R$. Then the Bifurcation from Simple Eigenvalues Theorem ([9, Theorem 1], [23,
Theorem 3.2.2]) says that in a neighborhood of $(1,\lambda_{k})$ the space of
solutions of $F(u,\lambda)=0$ is given by the path of trivial solutions
$(1,\lambda)$ and a path of nontrivial solutions $s\mapsto(u(s),\lambda(s))$
verifying: $\lambda(0)=\lambda_{k}$, $u(0)=1$, $\frac{\partial u}{\partial
s}(0)=P_{k}^{\alpha,\beta}$.

We will use ′ to denote differentiation with respect to the variable
$t\in[-1,1]$. We have

$(1-t^{2})u(s)^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)t)u(s)^{\prime}-\lambda(s)(u(s)-u(s)^{q})=0.$

Then differentiating with respect to $s$ we get:

$(1-t^{2})\left(\frac{du}{ds}\right)^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)t)\frac{du}{ds}^{\prime}-\lambda(s)\left(\frac{du}{ds}-qu(s)^{q-1}\frac{du}{ds}\right)-\frac{d\lambda}{ds}(u(s)-u(s)^{q})=0.$

And differentiating with respect to $s$ once more we get:

$(1-t^{2})\left(\frac{d^{2}u}{ds^{2}}\right)^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)t)\left(\frac{d^{2}u}{ds^{2}}\right)^{\prime}-\lambda(s)\left(\frac{d^{2}u}{ds^{2}}-qu(s)^{q-1}\frac{d^{2}u}{ds^{2}}-q(q-1)u(s)^{q-2}\left({\frac{du}{ds}}\right)^{2}\right)$

$-2\frac{d\lambda}{ds}\left(\frac{du}{ds}-qu(s)^{q-1}\frac{du}{ds}\right)-\frac{d^{2}\lambda}{ds^{2}}(u(s)-u(s)^{q})=0.$

Evaluating at $s=0$ we obtain:

$(1-t^{2})\left(\frac{d^{2}u}{ds^{2}}(0)\right)^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)t)\left(\frac{d^{2}u}{ds^{2}}(0)\right)^{\prime}$

$-\lambda_{k}\left((1-q)\frac{d^{2}u}{ds^{2}}(0)-q(q-1)(P_{k}^{\alpha,\beta})^{2}\right)-2\frac{d\lambda}{ds}(0)(1-q)P_{k}^{\alpha,\beta}=0.$

We let $w=\frac{d^{2}u}{ds^{2}}(0)$, multiply by
$P_{k}^{\alpha,\beta}(1-t)^{\alpha}(1+t)^{\beta}$ and integrate:

$\int_{-1}^{1}D_{u}F(1,\lambda_{k})[w]P_{k}^{\alpha,\beta}(1-t)^{\alpha}(1+t)^{\beta}dt+\lambda_{k}q(q-1)\int_{-1}^{1}(P_{k}^{\alpha,\beta})^{3}(1-t)^{\alpha}(1+t)^{\beta}dt$

$+2(q-1)\frac{d\lambda}{ds}(0)\int_{-1}^{1}(P_{k}^{\alpha,\beta})^{2}(1-t)^{\alpha}(1+t)^{\beta}dt=0$

Using Lemma 3.1, since $D_{u}F(1,\lambda_{k})[P_{k}^{\alpha,\beta}]=0$ we see
that the first term vanishes and then we obtain:

$\frac{d\lambda}{ds}(0)=\frac{-q\lambda_{k}\int_{-1}^{1}(P_{k}^{\alpha,\beta})^{3}(1-t)^{\alpha}(1+t)^{\beta}dt}{2\int_{-1}^{1}(P_{k}^{\alpha,\beta})^{2}(1-t)^{\alpha}(1+t)^{\beta}dt}.$

The last statement of the theorem then follows from Theorem 2.2.

∎

##  4\. Degenerate solutions

In this section we will prove Theorem 1.3. Fix the isoparametric function
$f^{d,c}$. We will assume that $c\leq 0$, as it was mentioned in the
introduction. Also we pick $\alpha,\beta$ by solving

$\beta-\alpha=(1/2)c$

$\alpha+\beta+2=\frac{n+d-1}{d},$

A $f^{d,c}$-invariant solution of Equation (1.1) is given by a function
$\varphi\in C^{2}[-1,1]$ which solves

(4.1)
$(1-t^{2})\varphi^{\prime\prime}(t)+(-(\alpha+\beta+2)t+\beta-\alpha)\varphi^{\prime}(t)-\frac{\lambda}{d^{2}}(\varphi(t)-\varphi^{q}(t))=0,$

where we point out that we have that $\alpha\geq\beta$ and $\alpha+\beta+1>0$.

We recall the following two results which appear in [3, Theorem 1.2, Theorem
4.1]:

######  Theorem 4.1.

If $1<q<q_{f}$ there exists $\lambda_{0}>0$ such that if $u$ is a positive
$f^{d,c}$-invariant solution of Equation (1.1) with $\lambda<\lambda_{0}$ then
$u=1$.

######  Theorem 4.2.

If $1<q<q_{f}$. For any positive numbers $0<\varepsilon<M$ we have that the
space of positive $f^{d,c}$-invariant solutions of Equation (1.1) with
$\lambda\in[\varepsilon,M]$ is compact.

Now we can prove Theorem 1.3.

###### Proof.

A $f^{d,c}$-invariant solution of Equation (1.1) is given by a solution $u$ of
Equation (4.1). Consider the space of nontrivial positive $f^{d,c}$-invariant
solutions of Equation (1.1):
$D=\\{(u,\lambda)\in(C^{2,\alpha}[-1,1]-\\{1\\})\times(0,\infty):u\;\text{is a
positive nontrivial solution of}\;(\ref{YamabeJacobi})\\}$. Consider the
bifurcation points $(1,d^{2}\lambda_{k})$ given by Theorem 3.2. Consider also
the path of nontrivial solutions $s\mapsto(u(s),\lambda(s))$,
$s\in(-\varepsilon,\varepsilon)$. Recall that we have that $\alpha\geq\beta$
and $\alpha+\beta+1>0$, as needed for Theorem 3.2. Since $u(0)=1$,
$\frac{du}{ds}(0)=P_{k}^{\alpha,\beta}$ and $P_{k}^{\alpha,\beta}$ has $k$
zeroes in $(-1,1)$. It follows that for $s$ close to zero, $s\neq 0$, the
solution $u(s)$ takes the value 1 exactly $k$-times.

Let $D_{k}^{+}$ be the connected component of $D$ containing the path
$(u(s),\lambda(s))$ with $s>0$ and $D_{k}^{-}$ be the connected component of
$D$ containing the path $(u(s),\lambda(s))$ with $s<0$. Let
$D_{k}=D_{k}^{+}\cup D_{k}^{-}$. Note that since $P_{k}^{\alpha,\beta}(1)>0$
we have that for $s>0$, $s$ small, $u(s)(1)>1$. On the other hand if at some
point $s>0$ we have that $u(s)(1)=1$ then we would have from Equation (3.1)
that $u$ is constant (equal to 1).

Let $(u,\lambda)\in D$. Then at any point $t_{0}$ such that $u(t_{0})=1$ we
have $u^{\prime}(t_{0})\neq 0$. Note also that $u(-1)\neq 1$ since $u$ must
satisfy 3.2. If $u(-1)>1$ ($<1$) then there exists a neighborhood $U$ of
$(u,\lambda)$ in $C^{2,\alpha}[-1,1]\times\mathbb{R}$ such that for any
$(v,\mu)\in U$ we have $v(-1)>1$ ($<1$) and $v$ and $u$ take the value 1 the
same number of times. It follows that on a connected subset $C$ of $D$ we have
for all $(u,\lambda)\in C$ the number of times $u$ takes the value $1$ is the
same, and also the sign of $u(-1)-1$. Therefore for all $(u,\lambda)\in D_{k}$
we have that $u^{-1}(1)$ has $k$ elements. It follows then from this
discussion that if $(1,\lambda)\in\overline{D_{k}}$ then
$\lambda=d^{2}\lambda_{k}$.

Theorem 3.2 says that $\frac{d\lambda}{ds}(0)<0$. Then for $s>0$ small, we
have that $u(s)$ is a solution of Equation (4.1) with
$\lambda=\lambda(s)<d^{2}\lambda_{k}$.

Now it follows from Theorem 4.1 that if $(u,\lambda)\in D_{k}^{+}$ then
$\lambda\geq\lambda_{0}$. From Theorem 4.2 we know that $\\{(u,\lambda)\in
D_{k}:\lambda\in[\lambda_{0},d^{2}\lambda_{k}]\\}$ is compact. Then there
exists $(u_{*},\lambda_{*})\in\overline{D_{k}^{+}}$ such that
$\lambda_{*}\leq\lambda$ for all $(u,\lambda)\in D_{k}^{+}$. But if $u_{*}$ is
constant then $(1,\lambda_{*})$ would be a bifurcation point and therefore
$\lambda_{*}=d^{2}\lambda_{i}$ for some $i<k$. But we already mentioned that
this cannot happen. Therefore $u_{*}$ is not constant and therefore
$(u_{*},\lambda_{*})\in D_{k}^{+}$.

If $D_{u}F(u_{*},\lambda_{*})$ were an isomorphism then in particular $DF$
would be surjective and we could apply the implicit function theorem. Note
that $D_{\lambda}F(u_{*},\lambda_{*})=u_{*}-u_{*}^{q}\neq 0$ and the kernel of
$DF$ is one-dimensional. Therefore there exists a regular path
$s\mapsto(u(s),\lambda(s))$ sucht that $F(u(s),\lambda(s))=0$, $u(0)=u_{*}$,
$\lambda(0)=\lambda_{*}$, $(u(s),\lambda(s)\in D_{k}^{+}$. Since $\lambda_{*}$
is the minimum value of $\lambda$ for $(u,\lambda)\in D_{k}^{*}$ we must have
that $\lambda^{\prime}(0)=0$. Then differentiating with respect to $s$ the
equation

$0=F(u(s),\lambda(s))=-\Delta u(s)+\lambda(s)(u(s)-u(s)^{q}),$

and evaluating at $s=0$, we obtain for $v=\frac{du}{ds}(0)\neq 0$,

$-\Delta v+\lambda_{*}(v-qu_{*}^{q-1}v)=0.$

This means that $v\in Kernel(D_{u}F(u_{*},\lambda_{*})$ which is a
contradiction. Therefore it is not true that $D_{u}F(u_{*},\lambda_{*})$ is an
isomorphism, which means that $u_{*}$ is a degenerate, $f^{d,c}$-invariant
solution of Equation (1.1). Note that if $(u,\lambda)\in D$ and
$t_{0}\in(-1,1)$ is a critical point of $u$ we have that if $u(t_{0})<1$ the
$t_{0}$ is a local minimum and if $u(t_{0})<1$ the $t_{0}$ is a local maximum.
It follows that the number of critical poins of $u\in D_{k}^{+}$ is exactly
$k-1$. Therefore using Remark 1.4, we have proved the theorem.

∎

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combined with translational invariance, allowed us to preserve the systems’
infinite length (that is, we were able to bypass finite “volume” artifacts).
In addition, we were able to preserve the infinite dimension of the local
Hilbert spaces for bosonic degrees of freedom. Whether this “Heisenberg
picture” approach will be easily generalizable to wMERA implementations for
interacting field theories remains to be seen; either way, efficient numerical
MERA algorithms in the “Schrödinger picture” have been developed for condensed
matter systems, and there is no reason to believe that they could not be
successfully adapted for (suitably regularized and discretized) relativistic
field theories.

Our motivation for proposing wMERA is obviously not to have an alternative
method for solving free field theories. The real goal is to use this approach
to explore new variational methods for solving challenging problems in quantum
field theory (such as strongly coupled dynamics) as well as the incorporation
of RG flow in quantum algorithms (to, e.g., improve their efficiency and
accuracy, including in the preparation of initial states, in the simulation of
real-time evolution, and in the readout and interpretation of the simulation’s
output.)

The next important step towards this goal is explore non-Gaussian entanglers
for interacting QFTs, and to understand their properties in different
dynamical regimes. Possible avenues would be to explore hybrid entangling
layers composed of the intercalation of ultra-local non-Guassian entanglers
with quasi-local Bogoliubov transformations, which parallels the perturbative
calculation approach; and, for systems that exhibit dualities or confinement,
to explore the introduction of _“ancillary”_ degrees of freedom to represent
the dual or confined states, and devise entanglers whose action is to map the
original states to the dual or confined states. Another important next step
would be to generalize the wMERA formulation to (3+1)d QFTs by exploring
optimal wavelet classes in 3 spatial dimensions. In this exploratory phase,
the wMERA results would have to be validated against standard perturbative
calculations in the weakly coupled regime, or against exact solutions, when
available.

###### Acknowledgements.

I thank Dan Carney, Lukasz Cincio, Burak Sahinoglu, and Varun Vaidya for
collaboration in very early stages of this project. I am also grateful to
Tanmoy Bhattacharya, Soonwon Choi, Rolando Somma, and Claudia Frugiuele for
helpful discussions. This work was supported by the program for Quantum
Information Science Enabled Discovery for High Energy Physics (QuantISED for
HEP), contract number KA2401032, under the U.S. Department of Energy Office of
Science. I also acknowledge support from the Quantum Science Center, one of
the National Quantum Information Science Research Centers supported by the
U.S. Department of Energy Office of Science. Los Alamos National Laboratory is
operated by Triad National Security, LLC, for the National Nuclear Security
Administration of the U.S. Department of Energy under contract number
89233218CNA000001. I thank the hospitality of the Aspen Center for Physics
(ACP) where this work was partially completed—the ACP is supported by the
National Science Foundation under grant PHY-1607611.

##  Appendix A Matrix exponentials of entangling parameters

In this appendix, we describe the approximations made in our numerical wMERA
algorithm in order to represent the exponential matrix function of the
entangling parameters $\zeta_{ij}$.

We begin by recalling that $\zeta_{ij}$ is a square matrix associated with an
infinite lattice system (that is, the integer indices $i,\,j$ run from
$-\infty$ to $+\infty$). However, due to discrete translational invariance,
$\zeta_{ij}$ obeys the restricted form:

$\zeta_{ij}=\zeta_{i-j}.$ (184)

This property enables the matrix $\zeta$ to be represented by an infinite
array of parameters $\zeta_{k}$ through the following decomposition:

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\zeta_{\,0}\begin{bmatrix}\ddots&0&~{}~{}~{}~{}&~{}~{}~{}~{}&~{}~{}~{}~{}\\\
0&1&0&&\\\ &0&1&0&\\\ &&0&1&0\vspace{-3pt}\\\
~{}~{}~{}~{}&~{}~{}~{}~{}&~{}~{}~{}~{}&0&\ddots\end{bmatrix}+\zeta_{\,1}\\!\begin{bmatrix}\ddots&1&~{}~{}~{}~{}&~{}~{}~{}~{}&~{}~{}~{}~{}\\\
0&0&1&&\\\ &0&0&1&\\\ &&0&0&1\vspace{-3pt}\\\
~{}~{}~{}~{}&~{}~{}~{}~{}&~{}~{}~{}~{}&0&\ddots\end{bmatrix}+\zeta_{-1}\\!\begin{bmatrix}\ddots&0&~{}~{}~{}~{}&~{}~{}~{}~{}&~{}~{}~{}~{}\\\
1&0&0&&\\\ &1&0&0&\\\ &&1&0&0\vspace{-3pt}\\\
~{}~{}~{}~{}&~{}~{}~{}~{}&~{}~{}~{}~{}&1&\ddots\end{bmatrix}\\!+\,\dots$ (185)
$\displaystyle\,=$
$\displaystyle~{}\sum_{k=-\infty}^{+\infty}\zeta_{k}\,\mathbb{M}^{(k)}\,,~{}~{}~{}~{}~{}\text{with}~{}~{}~{}~{}~{}\big{(}\mathbb{M}^{(k)}\big{)}_{ij}\,\equiv\,\begin{cases}\,1~{}~{}~{}\text{if}~{}~{}~{}j-i=k\\\
\,0~{}~{}~{}\text{otherwise.}\end{cases}$

Importantly, the matrices $\mathbb{M}^{(k)}$ satisfy the following relations:

$\big{[}\mathbb{M}^{(i)},\mathbb{M}^{(j)}\big{]}\,=\,0~{}~{}\forall~{}~{}i,\,j~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathbb{M}^{(i)}\mathbb{M}^{(j)}\,=\,\mathbb{M}^{(i+j)}.$
(186)

These relations allow us to trade the basis of matrices $\mathbb{M}^{(k)}$ for
powers of a commuting variable (a $c$-number) denoted by $m$. That is, in
(185), we make the replacement

$\mathbb{M}^{(k)}~{}~{}\longrightarrow~{}~{}~{}m^{k}.$ (187)

With (185) and (187), we then write the exponential matrix function of $\zeta$
as:

$\displaystyle
e^{\,\zeta}~{}=~{}\text{Exp}\left[\\!\\!\sum_{~{}~{}k=-\infty}^{+\infty}\\!\\!\\!\zeta_{k}\,m^{k}\right]~{}$
$\displaystyle=$
$\displaystyle~{}\prod_{k=-\infty}^{+\infty}e^{\,\zeta_{k}\,m^{k}}$ (188)
$\displaystyle=$
$\displaystyle~{}\left(\;\prod_{k=1}^{\infty}e^{\,\zeta_{-k}m^{-k}}\right)\times~{}e^{\,\zeta_{0}}~{}\times\left(\;\prod_{k=1}^{\infty}e^{\,\zeta_{k}\,m^{k}}\right).~{}~{}~{}~{}$

Defining
$\vec{\zeta}\equiv(\zeta_{1},\,\zeta_{2},\,\zeta_{3},\,\dots,\zeta_{n},\,\dots)$,
and recalling that the entangling matrix $\zeta_{ij}$ is symmetric for scalars
and anti-symmetric for fermions, we can then expand the product of
exponentials in (188) as:

$\displaystyle\left(\;\prod_{k=1}^{\infty}e^{\,\zeta_{k}\,m^{k}}\right)~{}$
$\displaystyle\equiv$
$\displaystyle~{}~{}\sum_{j=0}^{\infty}\,c_{j}(\vec{\zeta}\,)\,m^{j}$ (189)
$\displaystyle\left(\;\prod_{k=1}^{\infty}e^{\,\zeta_{-k}\,m^{-k}}\right)$
$\displaystyle=$
$\displaystyle~{}~{}\sum_{j=0}^{\infty}\,c_{j}(\pm\vec{\zeta}\,)\,m^{-j}~{}~{}~{}~{}\text{with}~{}~{}~{}~{}\begin{cases}\,c_{k}(+\vec{\zeta}\,)~{}~{}\text{for
scalars,}\\\ \,c_{k}(-\vec{\zeta}\,)~{}~{}\text{for fermions.}\end{cases}$
(190)

Note that expression (189) _defines_ the functions $c_{j}(\vec{\zeta}\,)$.
That is, we can obtain $c_{j}(\vec{\zeta}\,)$ by expanding the left-hand side
of (189) as a Taylor series in $m$ and selecting the coefficient multiplying
$m^{j}$.

Because of (184), the exponential matrix function of $\zeta$ also obeys
discrete translational invariance,

$\big{(}e^{\,\zeta\,}\big{)}_{ij}=\big{(}e^{\,\zeta\,}\big{)}_{i-j},$ (191)

and it can be expanded in powers of $m$,

$e^{\,\zeta\,}~{}=\sum_{~{}k=-\infty}^{+\infty}\big{(}e^{\,\zeta\,}\big{)}_{k}\,m^{k}\,.\\\
$ (192)

Using (188), (189), and (190), we can obtain the components of
$e^{\,\zeta\,}$. In particular, for a _scalar_ entangler:

$\big{(}e^{\,\zeta\,}\big{)}_{k}~{}=~{}\,e^{\,\zeta_{0}}~{}\sum_{j=0}^{\infty}~{}c_{j}(\vec{\zeta}\,)\,c_{j+|k|}(\vec{\zeta}\,)~{}~{}~{}~{}~{}~{}\text{(for~{}~{}$\zeta_{-i}=\zeta_{\,i}$)}.$
(193)

Similarly, for a _fermionic_ entangler:

$\big{(}e^{\,\zeta\,}\big{)}_{k}~{}=~{}\,e^{\,\zeta_{0}}\times\begin{cases}\;\sum_{j=0}^{\infty}~{}c_{k+j}(\vec{\zeta}\,)\,c_{j}(-\vec{\zeta}\,)~{}~{}~{}~{}\text{if}~{}~{}~{}~{}k\geq
0\\\ \vspace{-10pt}\\\
\;\sum_{j=0}^{\infty}~{}c_{j}(\vec{\zeta}\,)\,c_{j-k}(-\vec{\zeta}\,)~{}~{}~{}~{}\text{if}~{}~{}~{}~{}k<0\end{cases}~{}~{}~{}~{}\text{(for~{}~{}$\zeta_{-i}=-\zeta_{\,i}$)}.~{}~{}~{}~{}$
(194)

The definition of $c_{j}(\vec{\zeta}\,)$ in (189) as well as the expansions in
(193) and (194) are exact. In order to use these expansions in our numerical
algorithm, we are forced to make a few truncations:

  * •

As mentioned in subsection 5.1, at each wMERA layer we work with a finite
number of entangling parameters. That is, we replace the infinite array
$\vec{\zeta}$ by a finite one,

$\vec{\zeta}~{}=~{}(\zeta_{1},\,\zeta_{2},\,\zeta_{3},\,\dots,\zeta_{n}\,).$
(195)

  * •

We also set a maximum power $p$ for the power series expansion in any of the
$\zeta_{i}$ coefficients. That is, we make the approximation

$e^{\zeta_{i}m^{i}}~{}\approx~{}1+\zeta_{i\,}m^{i}+\frac{1}{2!}\big{(}\zeta_{i\,}m^{i}\big{)}^{2}+\dots+\frac{1}{p!}\big{(}\zeta_{i\,}m^{i}\big{)}^{p}.$
(196)

  * •

Finally, we truncate the number of $c_{j}(\vec{\zeta}\,)$ coefficients in the
expansion in (189). That is, we set

$\displaystyle
c_{j}(\vec{\zeta}\,)\,=\,0~{}~{}\text{for}~{}~{}j>j_{max}~{}~{}~{}~{}\Rightarrow~{}~{}~{}~{}\big{(}e^{\,\zeta\,}\big{)}_{j}\,=\,0~{}~{}\text{for}~{}~{}|j|>j_{max}.$
(197)

This set of truncations was not optimized in the numerical wMERA algorithm we
used in the examples discussed in section 5. Mapping out the optimal
truncations choices and how they affect the precision and accuracy of the
calculation is deferred to a future study.

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Table of large graphs with given degree and diameter

Francesc Comellas

Departament de Matemàtiques, Universitat Politècnica de Catalunya,

Barcelona, Catalonia, Spain<EMAIL_ADDRESS>
Abstract: We update the table of large undirected graphs with given degree and
diameter with results produced since the publication of the survey by M.
Miller and J. Širáň (Electron. J. Combin. DS14, 2nd. edition. May 2013).

Keywords: Large Graphs, Degree-Diameter Problem, Cayley Graphs
MSC: 05C12, 05C25, 05C35, 05C38, 05C76, 05C90, 90B10.

##  1 Introduction

The construction of graphs with the largest possible order for a given maximum
degree and diameter, known as the $(\Delta,D)$-problem, is a question of
particular interest in graph theory. This problem receives much attention due
to its implications in the design of topologies in interconnection networks
and its relevance to issues such as data alignment and the description of
cryptographic protocols. Additionally, the $(\Delta,D)$-problem relates to
various graph properties, including regularity, connectivity, and cycle
structure.

Hoffman and Singleton introduced the concept of Moore graphs [24], and proved
that the largest possible order of a graph with maximum degree $\Delta$
($\Delta>2$) and diameter $D$ is bounded by

$1+\Delta+\Delta(\Delta-1)+\dots+\Delta(\Delta-1)^{D-1}=\frac{\Delta(\Delta-1)^{D}-2}{\Delta-2}=n(\Delta,D)\vspace{-3mm}$

This value is called the Moore bound, and a graph attaining it is known as a
Moore graph. Different authors have proved that for $D\geq 2$ and $\Delta\geq
3$, there can only exist Moore graphs for $D=2$ and $\Delta=3,7\mbox{ and
}57$. For the first two cases the graphs are unique and are, respectively, the
Petersen’s graph on 10 vertices and the Hoffman–Singleton’s graph on 50
vertices. The existence of a Moore graph for $D=2$ and $\Delta=57$ is not
known. Thus, it is of interest to find graphs which for a given diameter and
maximum degree, have a number of vertices as close as possible to the Moore
bound.

The $(\Delta,D)$ problem for undirected graphs has been approached in
different ways. It is possible to give bounds to the order of the graphs for a
given maximum degree and diameter. On the other hand, as the theoretical
bounds are difficult to attain, most of the work deals with the construction
of graphs that, for this given diameter and maximum degree, have a number of
vertices as close as possible to the theoretical bounds.

Different techniques have been developed depending on the way graphs are
generated and their parameters are calculated.

Roughly half of the graphs in Table 1 correspond to Cayley graphs and have
been found by computer methods [25, 27]. The computer is used for generating
the graphs and testing for the desired properties.

Compounding is another technique that has proven useful in producing
$(\Delta,D)$ graphs. This technique generalizes a method introduced by
Quisquater [30], which involves replacing a single vertex in a bipartite Moore
graph with a complete graph. Gómez, Fiol and Serra [22] further modified this
technique to replace several vertices of a given graph with another graph or
copies of a graph, rearranging the edges appropriately (see also [9, 21, 20]).
Compound graphs have been a basic tool in constructing many large $(\Delta,D)$
graphs, particularly for small diameters.

Other large graphs have been found as graph products or from particular
methods. Table 1 gives the current state of the art for degrees up to sixteen
and diameter up to ten.

##  2 Recent results

Details for the following updates and their adjacency lists, and information
and adjacency lists for most graphs in Table 1 with order less than 20000, can
be downloaded from [8].

###  2.1 Addition of vertices

The $(13,3)=856$ new graph was obtained in 2021 by Vlad Pelekhaty [28] from
the graph $Q_{8}^{\prime}d$ with degree 13, diameter 3 and 851 vertices
described in [20] by adding five new vertices and reconnecting several
vertices. The resulting graph is regular, has girth 3 and average distance
2.818817.

###  2.2 Construction based on a symmetric graph.

The $(3,8)=360$ new graph was obtained in 2018 by Jianxiang Chen [7].

This graph is derived from the symmetric graph on 144 vertices with diameter 7
and girth 8 by a complete pairing of its edges as follows:
Let $G$ be the symmetric graph and $\sim$ the pairing relation on its edges,
see [8].
The new graph, $H$, is constructed as follows:
The vertex set of $H$ is $V(G)\cup E(G)$.
If $v\in V(G)$ and $u\in V(G)$, then they are not connected in $H$.
If $v\in V(G)$ and $u\in E(G)$, then they are connected in $H$ iff $v$ is
incident to $u$ in $G$.
If $v\in E(G)$ and $u\in E(G)$, then they are connected in $H$ iff $v\sim u$
by the pairing relation.
The graph $H$ is not a Cayley graph. It is regular with girth 13 and has
average distance 6.122563.

###  2.3 Semidirect product of cyclic groups

M. Abas obtained a Cayley graph $(16,2)=200$ by considering the semidirect
product $\mathbb{Z}_{10}\times\mathbb{Z}_{10})\rtimes\mathbb{Z}_{2}$ and
generating set $X=A\cup B\cup B^{-1}\cup C\cup C^{-1}$, where
$A=\\{(0,0,1)\\}$, $B=\\{(1,0,1),(1,3,1),(1,7,1),(5,0,1),(5,2,1)\\}$ and
$C=\\{(5,0,0),(4,1,0),(3,2,0)\\}$, see [1].

A Cayley graph $(9,8)=1697688$ from the semidirect product
$\mathbb{Z}_{72}\rtimes_{1413}\mathbb{Z}_{23579}$ with generating set
$\\{[8,5958],[27,6086],[37,22093],[33,22621],[36,2717]\\}$ was produced by A.
Rodriguez in 2013, see [31] .

The following table provides details of the additional new Cayley graph
entries in Table 1 obtained by the author from semidirect products of cyclic
groups.

$\begin{array}[]{|c|c|c|c|}\hline\cr\hline\cr(\Delta,D)&\mbox{ \bf
order}&\mbox{ \bf group }&\mbox{ \bf generators }\\\
\hline\cr\hline\cr(6,8)&76\,891&\mathbb{Z}_{17}\rtimes_{891}\mathbb{Z}_{4523}&\mbox{\footnotesize[6,1326],[4,1336],[14,1686]
}\\\
\cdashline{1-4}[2pt/2pt](7,6)&12\,264&\mathbb{Z}_{24}\rtimes_{90}\mathbb{Z}_{511}&\mbox{\footnotesize[13,77],[6,157],[15,50],[12,7]
}\\\
\cdashline{1-4}[2pt/2pt](7,7)&53\,020&\mathbb{Z}_{20}\rtimes_{729}\mathbb{Z}_{2651}&\mbox{\footnotesize[6,894],[17,2271],[18,2411],[10,1210]
}\\\
\cdashline{1-4}[2pt/2pt](8,5)&5\,115&\mathbb{Z}_{113}\rtimes_{390}\mathbb{Z}_{196}&\mbox{\footnotesize[13,277],[1,290],[4.21],[10,258].
}\\\
\cdashline{1-4}[2pt/2pt](9,4)&1\,640&\mathbb{Z}_{40}\rtimes_{24}\mathbb{Z}_{41}&\mbox{\footnotesize[25,28],[14,40],[29,11],[39,12],[20,35]}\\\
\cdashline{1-4}[2pt/2pt](10,4)&2\,331&\mathbb{Z}_{9}\rtimes_{44}\mathbb{Z}_{259}&\mbox{\footnotesize[8,132],[2,171],[2,71],[4,236],[6,240]}\\\
\cdashline{1-4}[2pt/2pt](10,5)&13\,203&\mathbb{Z}_{81}\rtimes_{22}\mathbb{Z}_{163}&\mbox{\footnotesize[49,70],
[64,134], [[78,95], [45,156], [14,90]}\\\
\cdashline{1-4}[2pt/2pt](11,5)&19\,620&\mathbb{Z}_{36}\rtimes_{434}\mathbb{Z}_{545}&\mbox{\footnotesize[22,21],
[30,484], [22,513], [33,116 ], [28,421], [18,285] }\\\
\cdashline{1-4}[2pt/2pt](12,5)&29\,621&\mathbb{Z}_{19}\rtimes_{1205}\mathbb{Z}_{1559}&\mbox{\footnotesize[4,358],
[15,963], [12,47], [9,233], [14,645], [12,1195]. }\\\
\cdashline{1-4}[2pt/2pt](13,5)&40\,488&\mathbb{Z}_{24}\rtimes_{362}\mathbb{Z}_{1687}&\mbox{\footnotesize[1,1454],
[5,1427], [2,1659], [15,837], [13,1606], [19,1105], [12,1029].}\\\
\cdashline{1-4}[2pt/2pt](14,5)&58\,095&\mathbb{Z}_{45}\rtimes_{191}\mathbb{Z}_{1291}&\mbox{\footnotesize[31,28],
[32,290], [28,326], [41,665], [18,278], [24,148], [36,259]. }\\\
\cdashline{1-4}[2pt/2pt](15,5)&77\,520&\mathbb{Z}_{48}\rtimes_{772}\mathbb{Z}_{1615}&\mbox{\footnotesize[3,482],[28,1131],[31,682],[47,1424],[2,831],[10,300],[23,1068],[24,0].
}\\\ \hline\cr\hline\cr\end{array}$

##  3 Table of the largest $(\Delta,D)$ graphs

D | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$\Delta$ |  |  |  |  |  |  |  |  |
3 |  P 10 |  $C_{5}\mathord{*}F_{4}$ 20 |  vC 38 |  vC 70 |  $Exoo$ 132 |  $Exoo$ 196 |  $Chen$ 360 |  $Exoo$ 600 |  $Conder$ 1 250
4 |  $K_{3}\mathord{*}C_{5}$ 15 |  Allwr 41 |  $Exoo$ 98 |  $H^{\prime}_{3}$ 364 |  $H_{3}(K_{3})$ 740 |  $Loz$ 1 320 |  $Loz$ 3 243 |  $Loz$ 7 575 |  $Loz$ 17 703
5 |  $K_{3}\mathord{*}X_{8}$ 24 |  $Exoo$ 72 |  $Exoo$ 212 |  $Loz$ 624 |  $H_{4}(K_{3})$ 2 772 |  $Loz$ 5 516 |  $Loz$ 17 030 |  $Loz$ 57 840 |  $Loz$ 187 056
6 |  $K_{4}\mathord{*}X_{8}$ 32 |  $Exoo$ 111 |  $Loz$ 390 |  $Loz$ 1 404 |  $H_{5}(K_{4})$ 7 917 |  $Loz$ 19 383 |  $Com$ 76 891 |  $Rod$ 331 387 |  $Loz$ 1 253 615
7 |  HS 50 |  $Exoo$ 168 |  $Sa$ 672 |  DH 2 756 |  $Com$ 12 264 |  $Com$ 53 020 |  $Loz$ 249 660 |  $Loz$ 1 223 050 |  $Loz$ 6 007 230
8 |  $P^{\prime}_{7}$ 57 |  $CM,Sa$ 253 |  $Loz$ 1 100 |  $Com$ 5 115 |  $H_{7}(K_{5})$ 39 672 |  $Loz$ 131 137 |  $Loz$ 734 820 |  $Loz$ 4 243 100 |  $Loz$ 24 897 161
9 |  $P^{\prime}_{8}d$ 74 |  $Q^{\prime}_{8}$ 585 |  $Com$ 1 640 |  $Rod$ 8 268 |  $H_{8}(K_{6})$ 75 893 |  $Loz$ 279 616 |  $Rod$ 1 697 688 |  $Loz$ 12 123 288 |  $Loz$ 65 866 350
10 |  $P^{\prime}_{9}$ 91 |  $Q^{\prime}_{8}d$ 650 |  $Com$ 2 331 |  $Com$ 13 203 |  $H_{9}(K_{6})$ 134 690 |  $Loz$ 583 083 |  $Loz$ 4 293 452 |  $Loz$ 27 997 191 |  $Loz$ 201 038 922
11 |  $Exoo$ 104 |  $Q^{\prime}_{8}d$ 715 |  $Q_{7}(T_{4})$ 3 200 |  $Com$ 19 620 |  $H_{7}(T_{4})$ 156 864 |  $Loz$ 1 001 268 |  $Loz$ 7 442 328 |  $Loz$ 72 933 102 |  $Loz$ 600 380 000
12 |  $P^{\prime}_{11}$ 133 |  $Q^{\prime}_{8}d^{+}$ 786 |  $Q^{\prime}_{8}\mathord{*}X_{8}$ 4 680 |  $Com$ 29 621 |  $H_{11}(K_{8})$ 359 772 |  $Loz$ 1 999 500 |  $Loz$ 15 924 326 |  $Loz$ 158 158 875 |  $Loz$ 1 506 252 500
13 |  MMS 162 |  $Pel$ 856 |  $Q_{9}(T_{4})$ 6 560 |  $Com$ 40 488 |  $H_{9}(T_{4})$ 531 440 |  $Loz$ 3 322 080 |  $Loz$ 29 927 790 |  $Loz$ 249 155 760 |  $Loz$ 3 077 200 700
14 |  $P^{\prime}_{13}$ 183 |  $Q^{\prime}_{8}d^{+}$ 916 |  $Q_{9}(T_{5})$ 8 200 |  $Com$ 58 095 |  $H_{13}(K_{10})$ 816 294 |  $K_{1}\Sigma_{8}H_{11}$ 6 200 460 |  $Loz$ 55 913 932 |  $Loz$ 600 123 780 |  $Loz$ 7 041 746 081
15 |  $P^{\prime}_{13}d$ 187 |  $(\otimes Q_{2,4})^{\prime}$ 1215 |  $Q_{11}(T_{4})$ 11 712 |  $Com$ 77 520 |  $H_{11}(T_{4})$ 1 417 248 |  $Loz$ 8 599 986 |  $Loz$ 90 001 236 |  $Loz$ 1 171 998 164 |  $Loz$ 10 012 349 898
16 |  $Abas$ 200 |  $(\otimes Q_{3})^{\prime}$ 1 600 |  $Q_{11}(T_{5})$ 14 640 |  $(\otimes H_{3})^{\prime}$ 132 496 |  $H_{11}(T_{5})$ 1 771 560 |  $K_{1}\Sigma_{8}H_{13}$ 14 882 658 |  $Loz$ 140 559 416 |  $Loz$ 2 025 125 476 |  $Loz$ 12 951 451 931

Table 1: Largest $(\Delta,D)$-graphs (June 2024). In bold new results since
the publication of the 2013 survey [27]. Detailed information and adjacency
lists for most of the graphs with order less than 20000 can be downloaded from
[8].

Graphs

Abas Graph found by M. Abas [1]

Allwr Special graphs found by J. Allwright [2]

Chen Graph found by J, Chen [7]

CM Cayley graph, $\mathbb{Z}_{M}\\!\\!\rtimes_{A}\\!\mathbb{Z}_{N}$, found by
F. Comellas and M. Mitjana

Com Cayley graphs, $\mathbb{Z}_{M}\\!\\!\rtimes_{A}\\!\mathbb{Z}_{N}$, found
by F. Comellas (this note)

Conder Graph found by M. Conder [10]

vC Compound graphs designed by C. von Conta [11]

$DH$ Cayley graph, $\mathbb{Z}_{M}\\!\\!\rtimes_{A}\\!\mathbb{Z}_{N}$, found
by M. J. Dinneen and P. Hafner [16]

$Exoo$ Graphs built by G. Exoo [17, 18].

$H_{q}$ Incidence graph of a regular generalized hexagon [3]

HS Hoffman-Singleton graph

$K_{n}$ Complete graph

$Loz$ Graphs built by E. Loz [25].

MMS Graph built by B. D. McKay, M. Miller, J. Širáň [26]

$P$ Petersen graph

Pel Graph found by V. Pelekhaty [28]

$P_{q}$ Incidence graph of projective plane [23]

$Q_{q}$ Incidence graph of a regular generalized quadrangle [3]

$Rod$ Cayley graphs,$\mathbb{Z}_{M}\\!\\!\rtimes_{A}\\!\mathbb{Z}_{N}$, found
by A. Rodriguez [31]

$Sa$ Cayley graphs,$\mathbb{Z}_{M}\\!\\!\rtimes_{A}\\!\mathbb{Z}_{N}$, found
by M. Sampels [32]

$T$ Tournament

Operations

$G*H$ Twisted product of graphs [4]

$Gd$ Duplication of some vertices of $G$ [12, 14]

$Gd^{+}$ Special duplication of vertices of $G$ [20]

$B^{\prime}$ Quotient of the bipartite graph $B$ by a polarity [12]

$\otimes B$ The component with polarity of the cartesian product of a
bipartite graph $B$ by itself [13]

$B(K)$ Substitution of vertices of a bipartite graph $B$ by complete graphs
$K$ [9, 19, 20, 29]

$B(T)$ Compound using a bipartite graph $B$ and a tournament $T$ [21]

$G\Sigma_{i}H$ Various compounding operations [22]

$\mathbb{Z}_{M}\\!\\!\rtimes_{A}\\!\mathbb{Z}_{N}$ Semidirect product of
cyclic groups [16]

Table 2: Graphs and operations of Table 1.

### Acknowledgements

We would like to acknowledge the invaluable contributions of Charles Delorme,
who sadly passed away in 2015. The format of this table and detailed
information on the graphs and graphs operations are based on his work.

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