Let
$\sharp \operatorname {hom}(G,H)$
denote the number of graph homomorphisms from a graph
${G=(V(G),E(G))}$
into an image graph
$H=(V(H),E(H))$
, that is,
$$ \begin{align*}\operatorname{hom}(G,H)=\{f:V(G)\to V(H) \mid (u,v)\in E(G) \Rightarrow (f(u),f(v))\in E(H)\}.\end{align*} $$
The following general inequality was proven by Sidorenko [Reference Sidorenko6].
Theorem 1. Let G denote any connected graph on
$k+1$
vertices and
$S_k$
the star graph with k edges. Then,
$\sharp \operatorname {hom}(G,H)\leq \sharp \operatorname {hom}(S_k,H) \text { for any graph }H.$
Define a partial order
$\preceq $
on the set
$\mathscr {G}_k$
of connected graphs on
$k+1$
vertices by declaring
$G'\succeq G$
if
$\sharp \operatorname {hom}(G',H)\geq \sharp \operatorname {hom}(G,H) \text { for every graph }H$
.
Aside from explicitly classifying the maximal elements of
$\preceq $
and its usefulness for counting problems in graphs, Sidorenko’s inequality is a valuable tool in various applications. For instance, if A is any positive symmetric
$n\times n$
matrix, Theorem 1 implies the inequality
$$ \begin{align*} \sum_{i,j\leq n}A^k_{i,j}\leq \sum_{i=1}^n\bigg(\sum_{j=1}^n A_{i,j}\bigg)^k, \end{align*} $$
which is due to Hoffman [Reference Hoffman3]. Further applications can be found in the context of noise sensitivity of Boolean functions [Reference Gopalan, Servedio, Wigderson and Raz2] and in the theory of complex networks, where Theorem 1 is central in deriving moment criteria for the continuity of clustering coefficients under local weak convergence [Reference Kurauskas4].
Three proofs of Theorem 1 can be found in the literature. Sidorenko’s original proof relies on the remarkable fact that the relation
$G\preceq G'$
is in one-to-one correspondence to an ordering of integral functionals on measure spaces which can be associated with G and
$G'$
. Certain combinatorial operations on graphs map to
$\log $
-convex combinations of these functionals for which Hölder-type inequalities hold. The corresponding inequalities for the homomorphism counts are then used to establish the extremality of star graphs and further relations between the homomorphism counts of concrete examples of graphs.
Csikvári and Lin [Reference Csikvári and Lin1] provide another proof of Theorem 1 that is close in spirit to Sidorenko’s work, but uses the Wiener index (the sum of all distances between pairs of vertices in a graph) and more elementary combinatorial operations on graphs to conclude the proof.
Finally, Levin and Peres [Reference Levin and Peres5] prove Theorem 1 by a brief and elegant probabilistic argument that connects the homomorphism count to the stationary distribution of the simple random walk on the target graph.
The aim of this note is to present a new and remarkably elementary proof of Sidorenko’s bound that relies solely on a short recursive enumeration argument and Hölder’s inequality on finite probability spaces.
Proof of Theorem 1.
Fix an arbitrary image graph
$H=(V(H),E(H))$
. Observe that removing edges from G can only increase
$\sharp \operatorname {hom}(G,H)$
; hence, it suffices to show
$$ \begin{align} \sharp \operatorname{hom}(T,H)\leq \sharp \operatorname{hom}(S_k,H) \end{align} $$
whenever T is any k-edge tree. Let
$\mathcal {T}(k,\ell )$
denote the set of k-edge trees with precisely
$\ell \leq k$
leaves. In particular,
$\mathcal {T}(k,k)=\{S_k\}$
. The bound (1) follows if we can show that
$$ \begin{align*} \max_{T\in \mathcal{T}(k,\ell)} \sharp \operatorname{hom}(T,H) \leq \max_{T\in \mathcal{T}(k,\ell+1)} \sharp \operatorname{hom}(T,H) \quad\text{for any }\ell<k. \end{align*} $$
To this end, we demonstrate that for every nonstar
$T\in \mathcal {T}(k,\ell )$
, there exists some k-edge tree
$T'$
with one more leaf that admits at least the same number of homomorphisms into H as T. Denote by
$\operatorname {sk}(T)$
the skeleton tree of T, obtained by removing all leaves from T. Since
$\ell <k$
,
$\operatorname {sk}(T)$
has at least two leaves. We designate the two leaves
$b_1,b_2$
of
$\operatorname {sk}(T)$
, and denote by
$T(b_1,b_2)$
the graph obtained from T by removing all leaves adjacent to
$b_1$
and
$b_2$
. We write
$\vec {d}_1,\vec {d}_2$
for the number of leaves removed at
$b_1$
and
$b_2$
, respectively. Calculating
$\sharp \operatorname {hom}(T,H)$
by first counting all maps of
$T(b_1,b_2)$
and then the possible choices for the images of the remaining leaves yields
$$ \begin{align} \sharp \operatorname{hom}(T,H) & = \sum_{u,v\in V(H)} \sharp \{f\kern1.5pt{\in}\kern1.5pt \operatorname{hom}(T(b_1,b_2),H): f(b_1)\kern1.5pt{=}\kern1.5pt u,f(b_2)\kern1.5pt{=}\kern1.5pt v \} \operatorname{deg}(u)^{\vec{d}_1}\operatorname{deg}(v)^{\vec{d}_2} \notag\\[-2pt] &=\sharp\operatorname{hom}(T(b_1,b_2),H) \sum_{u,v\in V(H)}p(u,v) \operatorname{deg}(u)^{\vec{d}_1}\operatorname{deg}(v)^{\vec{d}_2}, \end{align} $$
where
$p(u,v)$
denotes the probability that a uniformly chosen map in
$\operatorname {hom}(T(b_1,b_2),H)$
maps
$b_1$
to u and
$b_2$
to v. Denoting the marginals of
$p(\cdot ,\cdot )$
by
$p_1(\cdot )$
and
$p_2(\cdot )$
, we conclude with the help of Hölder’s inequality,
$$ \begin{align} \sharp \operatorname{hom}(T,H) & \leq \sharp\operatorname{hom}(T(b_1,b_2),H) \notag\\ &\quad \cdot \bigg( \sum_{u\in V(H)}p_1(u)\operatorname{deg}(u)^{\vec{d}_1+\vec{d}_2} \bigg)^{{\vec{d}_1}/{(\vec{d}_1+\vec{d}_2})} \bigg( \sum_{u\in V(H)}p_2(u)\operatorname{deg}(u)^{\vec{d}_1+\vec{d}_2}\bigg)^{{\vec{d}_2}/{(\vec{d}_1+\vec{d}_2})}. \end{align} $$
We assume, without loss of generality, that
$$ \begin{align*} \sum_{u\in V(H)}p_1(u)\operatorname{deg}(u)^{\vec{d}_1+\vec{d}_2}\geq \sum_{u\in V(H)}p_2(u)\operatorname{deg}(u)^{\vec{d}_1+\vec{d}_2}, \end{align*} $$
since if the opposite inequality holds, we may reverse the choice of
$b_1,b_2$
at the beginning. Thus, (3) leads to
$$ \begin{align} \sharp \operatorname{hom}(T,H) & \leq \sharp\operatorname{hom}(T(b_1,b_2),H) \sum_{u\in V(H)}p_1(u)\operatorname{deg}(u)^{\vec{d}_1+\vec{d}_2} \notag\\ & = \sum_{u\in V(H)}\sharp \{f\in \operatorname{hom}(T(b_1,b_2),H): f(b_1)=u \} \operatorname{deg}(u)^{\vec{d}_1+\vec{d}_2}. \end{align} $$
The last expression equals
$\sharp \operatorname {hom}(T',H)$
, where
$T'\in \mathcal {T}(k,\ell +1)$
is obtained from
$T(b_1,b_2)$
by attaching
$\vec {d}_1+\vec {d}_2$
leaves to
$b_1$
and consequently has precisely one more leaf than T.
In fact, the above line of reasoning also establishes a slightly stronger form of the statement that coincides with Sidorenko’s original formulation of the result with only a minor refinement.
Corollary 2 [Reference Sidorenko6, Theorem 1.2].
Let T denote any k-edge tree,
$S_k$
the star graph with k edges and
$T_{k-1,1}$
any tree obtained by attaching a single leaf to a leaf of
$S_{k-1}$
. Then,
$ T\preceq T_{k-1,1} \preceq S_k. $
Proof. All graphs in
$\mathcal {T}(k,k-1)$
have
$K_2$
as their skeleton tree. Since the latter graph is symmetric under swapping
$b_1$
and
$b_2$
, it follows that the distribution
$p(\cdot ,\cdot )$
used in (2) is symmetric. Let
$(U,V)$
denote a pair of random variables with distribution
$p(\cdot ,\cdot )$
. Then,
$$ \begin{align*} \sum_{u,v\in V(H)}p(u,v) \operatorname{deg}(u)^{\vec{d}_1}\operatorname{deg}(v)^{\vec{d}_2} & =\mathbb{E}[(\operatorname{deg}(U)^{\vec{d}_1+\vec{d}_2})^{{\vec{d}_1}/{(\vec{d}_1+\vec{d}_2})} (\operatorname{deg}(V)^{\vec{d}_1+\vec{d}_2})^{{\vec{d_2}}/{(\vec{d}_1+\vec{d}_2})} ] \\&=\mathbb{E}[e^{p_1 g(U)+(1-p_1)g(V)}] \end{align*} $$
for
$g(v)=(\vec {d}_1+\vec {d}_2)\log (\operatorname {deg}(v))$
and
$p_1={\vec {d}_1}/({\vec {d}_1+\vec {d}_2}).$
Since
$p(u,v)=p(v,u)$
, the map
$\phi :[0,1]\to [0,\infty )$
given by
$$ \begin{align*} \phi(p)=\mathbb{E}[{e}^{p g(U)+(1-p)g(V)}] \end{align*} $$
is symmetric. The proof of Theorem 1 shows that
$\phi $
attains its maxima at
$0$
and
$1$
. Furthermore,
$$ \begin{align*} \phi'(p)=\mathbb{E}[(g(U)-g(V)){e}^{p g(U)+(1-p)g(V)}, \quad \phi"(p)=\mathbb{E}[(g(U)-g(V))^2{e}^{p g(U)+(1-p)g(V)}], \end{align*} $$
and hence
$\phi $
is convex and attains its minimum at
$p=1/2$
. Consequently,
$\phi $
is nondecreasing on
$[1/2,1]$
, which implies that
$\mathcal {T}(k,k-1)$
is totally ordered with respect to
$\preceq $
, where the minimum is attained at
$\vec {d}_1=\vec {d}_2=(k-1)/2$
if k is odd and at
$\vec {d}_1=k/2, \vec {d}_2=k/2-1$
if k is even, and the maximum is attained at
$\vec {d}_1=k-2, \vec {d}_2=1$
.
We conclude this note by remarking that one can replace the use of Hölder’s inequality in the proof of Theorem 1 by an equally short argument using the weighted AM–GM inequality. It is an elementary analytic exercise to see that the two inequalities are in fact equivalent, so both variants of our argument are equally fundamental.
Alternative proof of Theorem 1 without Hölder’s inequality.
Employing the notation from the proof of Corollary 2, we see that
$$ \begin{align*} \phi(p)\leq p\mathbb{E}[{e}^{g(U)}]+(1-p)\mathbb{E}[{e}^{g(V)}]\leq \max\{\mathbb{E}[{e}^{g(U)}],\mathbb{E}[{e}^{g(V)}]\},\quad p\in[0,1], \end{align*} $$
where the first inequality follows from applying the weighted AM–GM inequality. The term on the right-hand side is precisely the term appearing on the right-hand side of (4).
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