Abstract matrix-tree theorem and Bernardi polynomial

TL;DR

This paper extends the matrix-tree theorem to directed and undirected graphs using a generalized Tutte polynomial, providing new identities and a shorter proof for the higher-degree case.

Contribution

It introduces a three-parameter family of identities involving a Tutte polynomial variant, generalizing the matrix-tree theorem with new proofs and parallel results for undirected graphs.

Findings

Proves a three-parameter family of identities for a Tutte polynomial variant.

Provides a shorter proof for the higher-degree matrix-tree theorem.

Establishes parallel results for undirected graphs.

Abstract

This paper is a continuation of arXiv:1612.03873. We prove a three-parameter family of identities (Theorem 1.1) involving a version of the Tutte polynomial for directed graphs introduced by Awan and Bernardi in arXiv:1610.01839. A particular case of this family (Corollary 1.6) is the higher-degree generalization of the matrix-tree theorem proved in arXiv:1612.03873, which thus receives a new proof, shorter (and less direct) than the original one. The theory has a parallel version for undirected graphs (Theorem 1.2).

Equations

$$B_G(q,y,z)=\sum _{f:\{1,\dots ,n\}\to \{1,\dots ,q\}}{y}^{\#f_G^{>}}{z}^{\#f_G^{<}}$$

$$C_G(q,y)=\sum _{f:\{1,\dots ,n\}\to \{1,\dots ,q\}}{y}^{\#f_G^{\ne }}$$

$${\mathcal{B}}_{n,k}(q,y,z)\stackrel{\text{mbox}def}{=}\sum _{G\in {\Gamma }_{n,k}}{B}_G(q,y,z)G.$$

$${\hat{\mathcal{B}}}_{n,k}(q,y,z)\stackrel{\text{mbox}def}{=}\sum _{G\in {\Gamma }_{n,k}}[B_G{]}_k(q,y,z)G.$$

$${\mathcal{Z}}_{n,k}(q,v)$$

$${\hat{\mathcal{Z}}}_{n,k}(q,v)$$

$$B_i(G)=\{\begin{array}{ll}G,& a\ne b,\\ -{\sum }_{m\ne a}{R}_{a,m;i}G,& a=b.\end{array}$$

$$\Delta {\mathcal{B}}_{n,k}(q,y,z)={\hat{\mathcal{B}}}_{n,k}(q,y-1,z-1).$$

$$\Delta {\mathcal{Z}}_{n,k}(q,v)=(-1{)}^k{\hat{\mathcal{Z}}}_{n,k}(q,-v).$$

$${\chi }_G^{\ge }(q)$$

$${\chi }_G^{>}(q)$$

$${\mathcal{X}}_{n,k}^{\ge }(q)\stackrel{\text{mbox}def}{=}{\mathcal{B}}_{n,k}(q,0,1)=\sum _{G\in {\Gamma }_{n,k}}{\chi }_G^{\ge }(q)G,$$

$${\mathcal{X}}_{n,k}^{>}(q)\stackrel{\text{mbox}def}{=}{\hat{\mathcal{B}}}_{n,k}(q,0,1)=\sum _{G\in {\Gamma }_{n,k}}{\chi }_G^{>}(q)G$$

$${\chi }_G^{\ge }(-1)$$

$${\chi }_G^{>}(-1)$$

$${\det }_{n,k}\stackrel{\text{mbox}def}{=}\frac{(-1{)}^k}{k!}{\chi }_G^{\ge }(-1)=\frac{(-1{)}^k}{k!}\sum _{G\in S_{n,k}}(-1{)}^{{\beta }_0(G)}G$$

$${\det }_{n,k}^I\stackrel{\text{mbox}def}{=}\frac{(-1{)}^k}{k!}\sum _{G\in S_{n,k}^I}(-1{)}^{{\beta }_0(G)}G\in {\mathcal{G}}_{n,k}$$

$$\Delta \sum _{I\subset \{1,\dots ,n\}}{\det }_{n,k}^I=\frac{(-1{)}^n}{k!}\sum _{I\subset \{1,\dots ,n\}}\sum _{G\in A_{n,k}^I}G.$$

$$y_{G,H}=(-1{)}^{\#\text{of loops in H}}{B}_H(q,y,z)=(-1{)}^{k-e(\Phi )}{B}_{\Phi }(q,y,z).$$

$$x_G=\sum _{\Phi \subseteq G}{y}_{G,L(\Phi )}=(-1{)}^k\sum _{\Phi \subseteq G}(-1{)}^{e(\Phi )}{B}_{\Phi }(q,y,z)=[B_G{]}_k(q,y-1,z-1).$$

$$x_G$$

$$=(-2{)}^k\sum _{\Psi \subseteq \Phi \subseteq G}{\left(-\frac{1}{2}\right)}^{e(\Phi )}{q}^{{\beta }_0(\Psi )}{v}^{e(\Psi )}=(-2{)}^k\sum _{\Psi \subseteq G}{q}^{{\beta }_0(\Psi )}{v}^{e(\Psi )}\sum _{\Phi \supseteq \Psi }{\left(-\frac{1}{2}\right)}^{e(\Phi )}$$

$$\displaystyle=(-2)^{k}\sum_{\Psi\subseteq G}q^{\beta_{0}(\Psi)}v^{e(\Psi)}\left(-\frac{1}{2}\right)^{e(\Psi)}\bigl{(}1-\frac{1}{2}\bigr{)}^{k-e(\Psi)}=(-1)^{k}Z_{G}(q,-v).$$

Full text

1. Definitions and main results

The following theory has two parallel versions — for directed and undirected graphs — so let us introduce notation for both cases.

By $\Gamma_{n,k}$ we denote the set of directed graphs with $n$ vertices numbered $1,\dots,n$ and $k$ edges numbered $1,\dots,k$; in other words, an element $G\in\Gamma_{n,k}$ is a $k$-element sequence $([a_{1},b_{1}],\dots,[a_{k},b_{k}])$ where $a_{1},\dots,a_{k},b_{1},\dots,b_{k}\in\{1,\dots,n\}$. Loops (edges $[a,a]$) and parallel edges (pairs $[a_{i},b_{i}]=[a_{j},b_{j}]$) are allowed. Similarly, by $\Upsilon_{n,k}$ we denote the set of unoriented graphs with $n$ numbered vertices and $k$ numbered edges: $\Upsilon_{n,k}=\{(\{a_{1},b_{1}\},\dots,\{a_{k},b_{k}\})\}$. The forgetful map $\vphantom{\cdot}\left|\smash{\cdot}\right|:\Gamma_{n,k}\to\Upsilon_{n,k}$ relates to every graph $G$ its undirected version $\vphantom{G}\left|\smash{G}\right|$ obtained by dropping the edge orientation: $[a,b]\mapsto\{a,b\}$.

Denote by $\mathcal{G}_{n,k}$ (resp., $\mathcal{Y}_{n,k}$) a vector space over ${\mathbb{C}}$ spanned by $\Gamma_{n,k}$ (resp., $\Upsilon_{n,k}$). The forgetful map is naturally extended to the linear map $\vphantom{\cdot}\left|\smash{\cdot}\right|:\mathcal{G}_{n,k}\to\mathcal{Y}_{n,k}$.

1.1. Main theorems

A Bernardi polynomial [1] is a map $B:\Gamma_{n,k}\to{\mathbb{Q}}[q,y,z]$ defined as

[TABLE]

where $f_{G}^{>}$ (resp., $f_{G}^{<}$) is the set of edges $[ab]$ of $G$ such that $f(b)>f(a)$ (resp., $f(b)<f(a)$). See [1] for a detailed analysis of the properties of $B_{G}$ (in particular, for the proof of its polynomiality).

Bernardi polynomial is a directed version of the full chromatic polynomial, which is a map $\mathop{\rm C}\nolimits:\Upsilon_{n,k}\to{\mathbb{Q}}[q,y]$ defined as

[TABLE]

where $f_{G}^{\neq}$ is the set of edges $[ab]$ of $G$ such that $f(b)\neq f(a)$. The classical Potts polynomial (as defined e.g. in [7]) is related to the full chromatic polynomial by $Z_{G}(q,v)=(v+1)^{k}\mathop{\rm C}\nolimits_{G}(q,1/(v+1))$; see [1] for details.

For any $G\in\Gamma_{n,k}$ (resp., $G\in\Upsilon_{n,k}$) we denote by $\widehat{G}$ the graph $G$ with all the loops deleted (and the numbering of the non-loop edges shifted accordingly); we have $\widehat{G}\in\Gamma_{n,k-\ell}$ (resp., $G\in\Upsilon_{n,k-\ell}$) where $\ell$ is the number of loops in $G$. It follows directly from the definition that $B_{G}(q,y,z)=B_{\widehat{G}}(q,y,z)$; in the undirected case $Z_{G}(q,v)=(v+1)^{\ell}Z_{\widehat{G}}(q,v)$.

The universal Bernardi polynomial is an element of $\mathcal{G}_{n,k}[q,y,z]$ defined as

[TABLE]

For a polynomial $P\in{\mathbb{C}}[q,y,z]$ denote by $[P]_{k}$ the sum of terms containing monomials $q^{s}y^{i}z^{j}$ with $i+j=k$ (and any $s$). The universal truncated Bernardi polynomial is an element of $\mathcal{G}_{n,k}[q,y,z]$ defined as

[TABLE]

Note that $[B_{G}]_{k}=0$ if (and only if) $G$ contains at least one loop; that is, $\widehat{{\mathcal{B}}}_{n,k}$ contains only loopless graphs. $\widehat{{\mathcal{B}}}_{n,k}$ is homogeneous of degree $k$ with respect to $y$ and $z$ and is not homogeneous with respect to $q$.

The universal Potts polynomial and the universal truncated Potts polynomial are elements of $\mathcal{Y}_{n,k}[q,v]$ defined, respectively, as

[TABLE]

For any $i=1,\dots,k$ and $p,q=1,\dots,n$ denote by $R_{p,q;i}:\Gamma_{n,k}\to\Gamma_{n,k}$ the map replacing the edge number $i$ of every graph $G\in\Gamma_{n,k}$ by the edge $[p,q]$ carrying the same number $i$. Also denote by $B_{i}:\mathcal{G}_{n,k}\to\mathcal{G}_{n,k}$ the linear operator acting on the graph $G\in\Gamma_{n,k}$ such that $[a,b]$ is its edge number $i$ as

[TABLE]

Following [2] call the product $\Delta\stackrel{{\scriptstyle\mbox{\scriptsize def}}}{{=}}B_{1}\dots B_{k}:\mathcal{G}_{n,k}\to\mathcal{G}_{n,k}$ the Laplace operator. The undirected version of the Laplace operator is defined as follows: if $G\in\Upsilon_{n,k}$ then $\Delta(G)\stackrel{{\scriptstyle\mbox{\scriptsize def}}}{{=}}\vphantom{\Delta(\Phi)}\left|\smash{\Delta(\Phi)}\right|$ where $\Phi\in\Gamma_{n,k}$ is any directed graph such that $G=\vphantom{\Phi}\left|\smash{\Phi}\right|$.

The main results of this paper are the following two theorems:

Theorem 1.1**.**

[TABLE]

and its undirected version

Theorem 1.2**.**

[TABLE]

1.2. Corollaries

1.2.1. Universal chrmomatic polynomials

Following [1], denote by $\chi_{G}^{{\geq}}$ (a chromatic polynomial of the directed graph $G\in\Gamma_{n,k}$) a polynomial such that for any $q=1,2,\dots$ the value $\chi_{G}^{{\geq}}(q)$ is equal to the number of mappings $f:\{1,\dots,n\}\to\{1,\dots,q\}$ such that $f(a)\geq f(b)$ for every edge $[ab]\in G$. Also denote by $\chi_{G}^{{>}}$ (a strict chromatic polynomial of $G$) a polynomial such that for any $q=1,2,\dots$ the value $\chi_{G}^{{>}}(q)$ is equal to the number of mappings $f:\{1,\dots,n\}\to\{1,\dots,q\}$ such that $f(a)>f(b)$ for every edge $[ab]\in G$.

Comparing these definitions with the definition of the Bernardi polynomial in Section 1.1 one obtains the equalities:

[TABLE]

Thus, one may call the elements of $\mathcal{G}_{n,k}$

[TABLE]

and

[TABLE]

universal chromatic polynomials. Substitution of $y=0$ and $z=1$ in Theorem 1.1 yields

Corollary 1.3**.**

$\Delta{\mathcal{X}}_{n,k}^{{\geq}}(q)=(-1)^{k}{\mathcal{X}}_{n,k}^{{>}}(q)$.

1.2.2. Higher matrix-tree theorems

A graph $G\in\Gamma_{n,k}$ is called acyclic if it contains no oriented cycles (in particular, no loops); $G$ is called totally cyclic (or strongly semiconnected, following the terminology of [2]) if every edge of $G$ is a part of a directed cycle.

It is possible to make further specialization of parameters in Corollary 1.3 due to the following

Proposition 1.4**.**

For any $G\in\Gamma_{n,k}$

[TABLE]

where $\beta_{0}(G)$ is the [math]-th Betti number (i.e. the number of connected components) of the graph $G$.

For proof see [1, Eq. (45) and Definition 5.1]. Note that it follows immediately from the definition that $\chi_{G}^{{>}}\equiv 0$ if (and only if) $G$ contains an oriented cycle (e.g. a loop), and that $\chi_{G}^{{\geq}}(q)=q^{\beta_{0}(G)}$ if $G$ is totally cyclic, so one half of each formula is evident (but not the other half).

Consider now (following [2]) the sum

[TABLE]

where by $\mathop{\rm\mathfrak{S}}\nolimits_{n,k}\subset\Gamma_{n,k}$ we denote the set of all totally cyclic graphs. Thus, Corollary 1.3 specializes to

Corollary 1.5**.**

$\Delta\det\nolimits_{n,k}=\frac{(-1)^{n}}{k!}\sum_{G\in\mathop{\rm\mathfrak{A}}\nolimits_{n,k}}G$.

where $\mathop{\rm\mathfrak{A}}\nolimits_{n,k}\subset\Gamma_{n,k}$ is the set of all acyclic graphs.

Corollary 1.5 admits a refinement. A totally cyclic graph may have isolated vertices (the ones not incident to any edge). Let $I=\{i_{1}<\dots<i_{s}\}\subset\{1,\dots,n\}$ be a set of vertices. We call a diagonal $I$-minor the element

[TABLE]

where $\mathop{\rm\mathfrak{S}}\nolimits_{n,k}^{I}$ is the set of all totally cyclic graphs $G\in\Gamma_{n,k}$ such that the vertices $i_{1},\dots,i_{s}$, and only they, are isolated. Similarly, denote by $\mathop{\rm\mathfrak{A}}\nolimits_{n,k}^{I}\subset\Gamma_{n,k}$ the set of all acyclic graphs such that $i_{1},\dots,i_{s}$, and only they, are sinks (vertices without edges starting at them); so Corollary 1.5 now looks like

[TABLE]

It follows from the definition of the Laplace operator that if $G\in\mathop{\rm\mathfrak{S}}\nolimits_{n,k}^{I}$ then $\Delta G=\sum_{H}x_{H}H$ where $x_{H}\in{\mathbb{Z}}$ and all $H$ have $i_{1},\dots,i_{s}$, and only them, as sinks. Since $\mathop{\rm\mathfrak{S}}\nolimits_{n,k}^{I}$ with different $I$ do not intersect, and the same is true for $\mathop{\rm\mathfrak{A}}\nolimits_{n,k}^{I}$, there is

Corollary 1.6** (of Corollary 1.5).**

For every $I=\{i_{1},\dots,i_{s}\}\subset\{1,\dots,n\}$ one has $\Delta\det\nolimits_{n,k}^{I}=\frac{(-1)^{n}}{k!}\sum_{G\in\mathop{\rm\mathfrak{A}}\nolimits^{I}_{n,k}}G$.

This is the abstract-matrix tree theorem [2, Theorem 1.7] which, in turn, is a higer-degree generalization of the celebrated matrix-tree theorem (first discovered by G. Kirchhoff in 1847 [3] and extended to the directed graphs by W. Tutte [6]).

Acknowledgements

The research was funded by the Russian Academic Excellence Project ‘5-100’ and by the grant No. 15-01-0031 “Hurwitz numbers and graph isomorphism” of the Scientific Fund of the Higher School of Economics.

2. Proofs

A graph $H\in\Gamma_{n,m}$ is called a subgraph of $G\in\Gamma_{n,k}$ (notation $H\subseteq G$) if it can be obtained from $G$ by deletion of several edges. (When one deletes the edge number $s$ from the graph, the numbers of the remaining edges are preserved if they are less than $s$ and are lowered by $1$ if they are greater than $s$.)

For convenience denote by $e(G)$ the number of edges of the graph $G$ (so $e(G)=k$ if $G\in\Gamma_{n,k}$. Proof of Theorem 1.1 involves the following well-known lemma:

Lemma 2.1** (Moebius inversion formula, [4]).**

Let $f:\bigcup_{k}\Gamma_{n,k}\to{\mathbb{C}}$ be a function on the set of graphs with $n$ vertices, and let the function $h$ on the same set be defined by the equality $h(G)=\sum_{H\subseteq G}f(H)$ for every $G\in\Gamma_{n,k}$. Then one has $(-1)^{e(G)}f(G)=\sum_{H\subseteq G}(-1)^{e(H)}h(H)$.

Proof of Theorem 1.1.

Let $\Delta{\mathcal{B}}_{n,k}(q,y,z)=\sum_{G\in\Gamma_{n,k}}x_{G}G$; by the definition of the Laplace operator $x_{G}\neq 0$ only if $G$ contains no loops. For a graph $H\in\Gamma_{n,k}$ the element $\Delta H\in\mathcal{G}_{n,k}$ contains a term $y_{G,H}G$ with $y_{G,H}\neq 0$ if and only if $\Phi\stackrel{{\scriptstyle\mbox{\scriptsize def}}}{{=}}\widehat{H}$ (the graph $H$ with all the loops deleted) is a subgraph of $G$. For any subgraph $\Phi\subseteq G$ of a loopless graph $G$ there exists exactly one $H\stackrel{{\scriptstyle\mbox{\scriptsize def}}}{{=}}L(\Phi)$ such that $\Phi=\widehat{H}$: every edge $[ab]$ present in $G$ but missing in $\Phi$ is replaced by the loop $[aa]$ in $H$.

Eventually, the coefficient $y_{G,H}$ in this case is

[TABLE]

where $\Phi=\widehat{H}$.

By [1, Eq. (21)] one has $\sum_{\Phi\subseteq G}[B_{\Phi}]_{e(\Phi)}(q,y-1,z-1)=B_{G}(q,y,z)$. Applying the Moebius inversion formula (Proposition 2.1) to this identity one obtains

[TABLE]

∎

Proof of Theorem 1.2.

is similar to that of Theorem 1.1: again, if $\Delta{\mathcal{Z}}_{n,k}(q,v)=\sum_{G}x_{G}G$ then $G$ entering the sum have no loops. A contribution $y_{G,H}$ of a graph $H$ that into $x_{G}$ is nonzero if and only if $\Phi=\widehat{H}$ is a subgraph of $G$. For a subgraph $\Phi\subseteq G$ having $e(\Phi)$ edges there are $2^{k-e(\Phi)}$ graphs $H$ such that $\Phi=\widehat{H}$: every edge $[ab]$ present in $G$ but missing in $\Phi$ may correspond either to a loop $[aa]$ or to a loop $[bb]$ in $H$; recall that $a\neq b$ because $G$ is loopless.

The contribution $y_{G,H}$ of all such graphs $H$ into $x_{G}$ is the same and is equal to $(-1)^{k-e(\Phi)}Z_{\Phi}(q,v)$. Now by [5] one has $Z_{G}(q,v)=\sum_{H\subseteq G}q^{\beta_{0}(H)}v^{e(H)}$, and therefore

[TABLE]

∎