On an Ordering-Dependent Generalization of the Tutte Polynomial

Abstract

A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction–deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte–Fortuin–Kasteleyn polynomial is recovered.

Appendices

Appendix A: Proof of Lemma 2

Let us recall Lemma 2:

Lemma 3

In notations of Theorem 21, for all $k > 1$,

$$\begin{aligned} C_{k1;B}(G;\epsilon ,\epsilon ') = \epsilon ^{1}_B \sum _{\ell >1}^kA^{(0)}_{\ell 1} C_{k\ell ;B}(G;\epsilon ,\epsilon ') . \end{aligned}$$

(A.1)

Working at fixed graphs B and G, we simplify the notations as follows: $C_{k\ell ;B}(G;\epsilon ,\epsilon ')=C_{k\ell }$. We prove the relation (A.1) by recurrence on k. At $k=2$, we have $C_{21}= \epsilon ^{1}_B A^{(0)}_{21} = \epsilon ^{1}_B A^{(0)}_{21} C_{22}$. Let us assume the result holds at k. Calculating the r.h.s of (A.1) at the next order $k+1$, we find

$$\begin{aligned} \epsilon ^{1}_B \sum _{\ell>1}^{k+1}A^{(0)}_{\ell 1} C_{(k+1)\ell }= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \epsilon ^{1}_BA^{(0)}_{k1}C_{(k+1)k} + \epsilon ^{1}_BA^{(0)}_{(k-1)1}C_{(k+1)(k-1)}\nonumber \\&+ \,\epsilon ^{1}_B \sum _{\ell>1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=0}^{(k+1)-\ell -1} \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , (k+1)-1\}}} \mathcal {A}_{(k+1)\ell ;B} (Q_p)\nonumber \\= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \epsilon ^{1}_BA^{(0)}_{k1}C_{(k+1)k} + \epsilon ^{1}_BA^{(0)}_{(k-1)1}C_{(k+1)(k-1)}\nonumber \\&+\, \epsilon ^{1}_B \sum _{\ell>1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell } \left[ \sum _{\mathop { |Q_p|=p \;\text {and}\; k\in Q_p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k\}}} + \sum _{\mathop { |Q_p|=p \; \text {and}\; k\notin Q_p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k\}}} \right] \mathcal {A}_{(k+1)\ell ;B} (Q_p) \nonumber \\= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \epsilon ^{1}_BA^{(0)}_{k1}C_{(k+1)k} + \epsilon ^{1}_BA^{(0)}_{(k-1)1}C_{(k+1)(k-1)}\nonumber \\&+\, \epsilon ^{1}_B \sum _{\ell >1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell } \left[ \sum _{\mathop { |Q_p|=p \;\text {and}\; k\in Q_p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k\}}} + \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-1\}}} \right] \mathcal {A}_{(k+1)\ell ;B} (Q_p).\nonumber \\ \end{aligned}$$

(A.2)

We expand $C_{(k+1)(k-1)}$ as

$$\begin{aligned} C_{(k+1)(k-1)} = \mathcal {A}_{(k+1)(k-1)}(Q_0)+ \mathcal {A}_{(k+1)(k-1)}(\{k\}) = \epsilon ^{k-1}_B (A^{(k-2)}_{(k+1)(k-1)} + \epsilon ^{k}_B A^{(k-1)}_{(k+1)k}A^{(k-2)}_{k(k-1)}). \end{aligned}$$

(A.3)

For the terms in (A.2) such that $k \in Q_p$, which implies $p\ge 1$, because $j_{p}^Q\le k$, we must have $j^Q_p=k$ and so we can write

$$\begin{aligned} \mathcal {A}_{(k+1)\ell ;B} (Q_p) = \epsilon _B^{k}A^{(k-1)}_{(k+1) k} \left( \epsilon ^\ell _B \prod _{a=1}^{p-1} \epsilon ^{j^Q_a}_B \right) \left[ A^{(j^Q_{p-1}-1)}_{k j^Q_{p-1}}\left( \prod _{i=2}^{p-1} A^{(j^Q_{i-1}-1)}_{j^Q_i j^Q_{i-1}} \right) A^{(\ell -1)}_{ j^Q_1 \ell } \right] . \end{aligned}$$

(A.4)

The sum over the terms in (A.2) with $k\in Q_{p\ge 1}$ yields, after adjusting the variable $p-1 \rightarrow p$,

$$\begin{aligned}&\epsilon ^{1}_BA^{(0)}_{k1}C_{(k+1)k} + \epsilon ^{k}_B\epsilon ^{1}_BA^{(0)}_{(k-1)1} A^{(k-1)}_{(k+1)k}A^{(k-2)}_{k(k-1)} \nonumber \\&\quad + \,\epsilon ^{1}_B \sum _{\ell>1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=1}^{k-\ell } \left[ \sum _{\mathop { |Q_p|=p \;\text {and}\; k\in Q_p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k\}}} \right] \epsilon _B^{k}A^{(k-1)}_{(k+1) k} \left( \epsilon ^\ell _B \prod _{a=1}^{p-1} \epsilon ^{j^Q_a}_B \right) \nonumber \\&\quad \times \left[ A^{(j^Q_{p-1}-1)}_{k j^Q_{p-1}}\left( \prod _{i=2}^{p-1} A^{(j^Q_{i-1}-1)}_{j^Q_i j^Q_{i-1}} \right) A^{(\ell -1)}_{ j^Q_1 \ell } \right] \nonumber \\&= \epsilon ^{1}_BA^{(0)}_{k1} \epsilon ^{k}_B A^{(k-1)}_{(k+1)k} + \epsilon ^{k}_B\epsilon ^{1}_BA^{(0)}_{(k-1)1} A^{(k-1)}_{(k+1)k}A^{(k-2)}_{k(k-1)} \nonumber \\&\quad +\, \epsilon _B^{k}A^{(k-1)}_{(k+1) k}\; \epsilon ^{1}_B \sum _{\ell>1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=1}^{k-\ell } \left[ \sum _{\mathop { |Q_p|=p -1 }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-1\}}} \right] \left( \epsilon ^\ell _B \prod _{a=1}^{p-1} \epsilon ^{j^Q_a}_B \right) \nonumber \\&\quad \times \left[ A^{(j^Q_{p-1}-1)}_{k j^Q_{p-1}}\left( \prod _{i=2}^{p-1} A^{(j^Q_{i-1}-1)}_{j^Q_i j^Q_{i-1}} \right) A^{(\ell -1)}_{ j^Q_1 \ell } \right] \nonumber \\&= \epsilon ^{1}_BA^{(0)}_{k1} \epsilon ^{k}_B A^{(k-1)}_{(k+1)k} + \epsilon ^{k}_B \epsilon ^{1}_BA^{(0)}_{(k-1)1} A^{(k-1)}_{(k+1)k}A^{(k-2)}_{k(k-1)} \nonumber \\&\quad + \epsilon _B^{k}A^{(k-1)}_{(k+1) k}\; \epsilon ^{1}_B \sum _{\ell>1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell -1} \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-1\}}} \mathcal {A}_{k\ell ;B} (Q_p) = \epsilon _B^{k}A^{(k-1)}_{(k+1) k}\; \epsilon ^{1}_B \sum _{l>1}^{k}A^{(0)}_{\ell 1} C_{k\ell } \nonumber \\&=\epsilon _B^{k}A^{(k-1)}_{(k+1) k}\; C_{k1}, \end{aligned}$$

(A.5)

where in the last line, use has been made of the recurrence hypothesis. Let us concentrate on the second type of terms $k\notin Q_p$ in (A.2) that we write, because there is no subsets of size $k-\ell >0$ (omitting $\epsilon ^{1}_BA^{(0)}_{(k+1)1}$)

$$\begin{aligned}&\epsilon ^{k-1}_B \epsilon ^{1}_BA^{(0)}_{(k-1)1} A^{(k-2)}_{(k+1)(k-1)} + \epsilon ^{1}_B \sum _{\ell>1}^{k-2}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell -1} \left[ \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-1\}}} \right] \mathcal {A}_{(k+1)\ell ;B} (Q_p) \nonumber \\&= \epsilon ^{k-1}_B \epsilon ^{1}_BA^{(0)}_{(k-1)1} A^{(k-2)}_{(k+1)(k-1)} + \epsilon ^{1}_B A^{(0)}_{(k-2) 1} (\mathcal {A}_{(k+1)(k-2);B} (Q_0)+ \mathcal {A}_{(k+1)(k-2);B} (\{k-1\})) \nonumber \\&\quad +\, \epsilon ^{1}_B \sum _{\ell>1}^{k-3}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell -1} \left[ \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-1\}}} \right] \mathcal {A}_{(k+1)\ell ;B} (Q_p) \nonumber \\&= \epsilon ^{k-1}_B \epsilon ^{1}_BA^{(0)}_{(k-1)1} A^{(k-2)}_{(k+1)(k-1)} + \epsilon ^{1}_B A^{(0)}_{(k-2) 1} \left( \epsilon _B^{k-2} A^{(k-1)}_{(k+1)(k-2)} \right. \nonumber \\&\quad \left. +\, \epsilon ^{k-2}_B \epsilon ^{k-1}_B A^{(k-2)}_{(k+1)(k-1)} A^{(k-3)}_{(k-1)(k-2)} \right) \nonumber \\&\quad +\, \epsilon ^{1}_B \sum _{\ell >1}^{k-3}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell -1} \left[ \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-1\}}} \right] \mathcal {A}_{(k+1)\ell ;B} (Q_p) . \end{aligned}$$

(A.6)

Now we iterate the same procedure and split the sum over subsets $Q_p \subset \{\ell +1,\dots , k-1\}$, among the terms containing $k-1$ and those which do not. Using the same decomposition as in (A.4) followed by the expansion (A.5), we obtain by summing those terms containing $k-1$, and a few algebra from (A.6):

$$\begin{aligned}&\epsilon ^{k-1}_B A^{(k-2)}_{(k+1)(k-1)}\left[ \epsilon ^{1}_BA^{(0)}_{(k-1)1} + \epsilon ^{1}_B\epsilon ^{k-2}_B A^{(0)}_{(k-2) 1} A^{(k-3)}_{(k-1)(k-2)} \right. \nonumber \\&\left. + \epsilon ^{1}_B \sum _{\ell>1}^{k-3}A^{(0)}_{\ell 1} \sum _{p=0}^{k-\ell -2} \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{\ell +1,\dots , k-2\}}} \mathcal {A}_{(k-1)\ell ;B} \right] \nonumber \\&= \epsilon ^{k-1}_B A^{(k-2)}_{(k+1)(k-1)}\Big [ \epsilon ^{1}_BA^{(0)}_{(k-1)1} C_{(k-1)(k-1)} + \epsilon ^{1}_BA^{(0)}_{(k-2)1} C_{(k-1)(k-2)} + \epsilon ^{1}_B \sum _{\ell>1}^{k-3}A^{(0)}_{\ell 1}C_{(k-1)\ell } \Big ]\nonumber \\&= \epsilon ^{k-1}_B A^{(k-2)}_{(k+1)(k-1)}\Big [ \epsilon ^{1}_B \sum _{\ell >1}^{k-1} A^{(0)}_{\ell 1} C_{(k-1)\ell } \Big ] = \epsilon ^{k-1}_B A^{(k-2)}_{(k+1)(k-1)} C_{(k-1)1}, \end{aligned}$$

(A.7)

where again we have used the recurrence hypothesis, this time, at order $k-1$. The procedure can be pursued until no more terms are left in the sum, and one gets as an upshot

$$\begin{aligned} \epsilon ^{1}_B \sum _{\ell >1}^{k+1}A^{(0)}_{\ell 1} C_{(k+1)\ell }= & {} \epsilon _B^{k}A^{(k-1)}_{(k+1) k}\; C_{k1} + \epsilon ^{k-1}_B A^{(k-2)}_{(k+1)(k-1)} C_{(k-1)1} + \dots \nonumber \\&+ \,\epsilon ^{2}_B A^{(1)}_{(k+1)2} C_{21} +\epsilon ^{1}_BA^{(0)}_{(k+1)1} C_{11} \nonumber \\= & {} \sum _{\ell =1}^{k} \epsilon _B^{\ell }A^{(\ell -1)}_{(k+1) \ell }\; C_{\ell 1} . \end{aligned}$$

(A.8)

We substitute the value of $C_{\ell 1}$ in the above expression and obtain

$$\begin{aligned} \epsilon ^{1}_B \sum _{\ell >1}^{k+1}A^{(0)}_{\ell 1} C_{(k+1)\ell }= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \sum _{\ell =2}^{k} \; \sum _{p=0}^{\ell -2} \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{2,\dots , \ell -1\}}} \epsilon _B^{\ell }A^{(\ell -1)}_{(k+1) \ell }\, \mathcal {A}_{\ell 1;B} (Q_p) \nonumber \\= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \sum _{p=0}^{k-2} \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{2,\dots , k -1\}}} \sum _{\ell =\max (p+2,j^Q_{p}+1)}^{k} \epsilon _B^{\ell }A^{(\ell -1)}_{(k+1) \ell }\, \mathcal {A}_{\ell 1;B} (Q_p) .\nonumber \\ \end{aligned}$$

(A.9)

Observing that $2\le j^Q_1< \dots < j^Q_p$, then $j^Q_p \ge p+2$ such that $\max (p+2,j^Q_{p}+1)=j^Q_{p}+1$. Furthermore, because $j^Q_p < \ell$, we re-express the above formula as

$$\begin{aligned} \epsilon ^{1}_B \sum _{\ell >1}^{k+1}A^{(0)}_{\ell 1} C_{(k+1)\ell }= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \sum _{p=0}^{k-2} \sum _{\mathop { |Q_p|=p }\limits ^{Q_p \subseteq \{2,\dots , k -1\}}} \sum _{\ell =j^Q_{p}+1}^{k} \mathcal {A}_{(k+1)1;B} (Q_p \cup \{\ell \}) \nonumber \\= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \sum _{p=0}^{k-2} \sum _{\mathop { |Q_{p+1}|=p+1 }\limits ^{Q_{p+1} \subseteq \{2,\dots , k\}}} \mathcal {A}_{(k+1)1;B} (Q_{p+1}) \nonumber \\= & {} \epsilon ^{1}_BA^{(0)}_{(k+1)1} + \sum _{p=1}^{k-1} \sum _{\mathop { |Q_{p}|=p }\limits ^{Q_{p} \subseteq \{2,\dots , k\}}} \mathcal {A}_{(k+1)1;B} (Q_{p}) \nonumber \\= & {} \sum _{p=0}^{k-1} \sum _{\mathop { |Q_{p}|=p }\limits ^{Q_{p} \subseteq \{2,\dots , k\}}} \mathcal {A}_{(k+1)1;B} (Q_{p}) = C_{(k+1)1}. \end{aligned}$$

(A.10)

$\square$

Appendix B: A Worked Out Example

Consider the graph G of Fig. 3, with edge set $\{e_{i}\}_{i=1,\dots , 5}$. We call $\{e_1,e_3,e_5\}=E_1$ the edge set of $G_1$, and $\{e_2,e_4\} = E_2$, the edge set of $G_2$. Note that the subgraphs $G_1$ and $G_2$ are symmetric under relabelling, so it does not matter to claim which edge is which in that figure.

Let us pick the sequence $(e_1,e_2,e_3,e_4,e_5)$ in which we want to perform the contraction deletion.

Fig. 3

A disconnected graph $G=G_1 \cup G_2$

Now, we can apply the state sum to find the polynomial. We will use a graphical representation to compute the $2^5$ terms of the sum and will simplify the notations. We have

(B.11)

Let us concentrate on the first term. Because the subgraph contains all edges, we can write (working at fixed subgraph B, we omit it in the notations):

(B.12)

with

$$\begin{aligned} \widehat{\lambda }_{1}= & {} \lambda _1\,; \nonumber \\ \widehat{\lambda }_{2}= & {} C_{2 1;B}(G;\epsilon ,\epsilon ')\lambda _1 +\lambda _2 = \epsilon \Big (A^{(0)}_{21} = 0 \Big ) \lambda _1 +\lambda _2 = \lambda _2 \,; \nonumber \\ \widehat{\lambda }_{3}= & {} C_{3 1}(G;\epsilon ,\epsilon ')\lambda _1 +C_{32}(G;\epsilon ,\epsilon ')\lambda _2 + \lambda _3 = C_{3 1}(G;\epsilon ,\epsilon ')\lambda _1 +\epsilon \Big (A^{(1)}_{32} = 0\Big )\lambda _2 + \lambda _3 \nonumber \\= & {} \epsilon \Big (A^{(0)}_{31}=1\Big ) \lambda _1 + \lambda _3 = \epsilon \lambda _1 + \lambda _3, \nonumber \\ \text{ where }&C_{3 1}(G;\epsilon ,\epsilon ') = \mathcal {A}_{31} (\emptyset ) + \mathcal {A}_{31} (\{2\}) = \epsilon A^{(0)}_{31} + \epsilon ^2\Big (A^{(1)}_{32}=0\Big )\Big (A^{(0)}_{21} = 0\Big ) = \epsilon A^{(0)}_{31}\,; \nonumber \\ \widehat{\lambda }_{4}= & {} (C_{41}(G;\epsilon ,\epsilon ')=0)\lambda _1 + \epsilon (A^{(1)}_{42}=1) \lambda _2 + \epsilon (A^{0}_{43}=0)\lambda _3 + \lambda _4 = \epsilon \lambda _2 + \lambda _4 , \nonumber \\ \text{ where }&C_{41}(G;\epsilon ,\epsilon ') = \mathcal {A}_{41} (\emptyset ) + \mathcal {A}_{41} (\{2\}) + \mathcal {A}_{41} (\{3\}) \nonumber \\&= \epsilon (A^{(0)}_{41}=0) + \epsilon ^2 A^{(1)}_{42}(A^{(0)}_{21}=0) + \epsilon ^2 (A^{(2)}_{43}=0)A^{(0)}_{31} =0 \nonumber \\ \text{ and }&C_{42}(G;\epsilon ,\epsilon ') = \mathcal {A}_{42} (\emptyset ) + \mathcal {A}_{43} (\{3\}) = \epsilon A^{(1)}_{42} + \epsilon ^2 (A^{(2)}_{43}=0)(A^{(1)}_{32} =0)= \epsilon A^{(1)}_{42}\,; \nonumber \\ \widehat{\lambda }_{5}= & {} C_{51}(G;\epsilon ,\epsilon ')\lambda _1 + (C_{52}(G;\epsilon ,\epsilon ')=0)\lambda _2 + C_{53}(G;\epsilon ,\epsilon ')\lambda _3 + \epsilon (A^{(3)}_{54}=0)\lambda _4 + \lambda _5 \nonumber \\= & {} (\epsilon + \epsilon ^2)\lambda _1 + \epsilon \lambda _3 + \lambda _5 , \nonumber \\ \text{ where }&C_{51}(G;\epsilon ,\epsilon ') = \mathcal {A}_{51} (\emptyset ) + \mathcal {A}_{51} (\{2,3,4\}) \nonumber \\&+ \,\mathcal {A}_{51} (\{2\}) + \mathcal {A}_{51} (\{3\}) +\mathcal {A}_{51} (\{4\}) + \mathcal {A}_{51} (\{2,3\}) + \mathcal {A}_{51} (\{2,4\}) + \mathcal {A}_{51} (\{3,4\}) \nonumber \\&= \epsilon A^{(0)}_{51} + (\epsilon ^{4}A^{(3)}_{54}...A^{(0)}_{21}=0) + \epsilon ^2 (A^{(1)}_{52}A^{(0)}_{21}=0) + \epsilon ^2 (A^{(2)}_{53}A^{(0)}_{31}=1)\nonumber \\&+ \,\epsilon ^2 (A^{(4)}_{54}A^{(0)}_{41}=0) + (\mathcal {A}_{51} (\{2,3\}) =0) + (\mathcal {A}_{51} (\{2,4\}) =0) + (\mathcal {A}_{51} (\{3,4\}) =0) \nonumber \\&= \epsilon A^{(0)}_{51} + \epsilon ^2 (A^{(2)}_{53}A^{(0)}_{31}=1) = \epsilon + \epsilon ^2 \nonumber \\ \text{ and }&C_{53}(G;\epsilon ,\epsilon ') = \mathcal {A}_{53} (\emptyset ) + (\mathcal {A}_{53} (\{4\}) =0) = \epsilon A^{(2)}_{53} , \end{aligned}$$

(B.13)

as

(B.14)

where in the two last equalities, we emphasize the factorization of this monomial between the contribution of two subgraphs of $G_1$ and $G_2$. Using the same technique, we list

(B.15)

We can already read that the polynomial associated with the graph $G_2$ is given by

$$\begin{aligned} P^{\epsilon ,\epsilon '}(G_2;q;\{\lambda _j\}_{j=2,4})= & {} q\Big [\alpha (\lambda _2) \alpha (\lambda _4 + \epsilon \lambda _2) + \beta (\lambda _2) \alpha (\lambda _4 + \epsilon ' \lambda _2) \nonumber \\&+\, \alpha (\lambda _2) \beta (\lambda _4 + \epsilon \lambda _2) + q \beta (\lambda _2) \beta (\lambda _4 + \epsilon ' \lambda _2)\Big ] . \end{aligned}$$

(B.16)

Now we can calculate the following terms

(B.17)

We deal with the next type of terms of the form:

(B.18)

Finally, we evaluate

(B.19)

Adding all contributions, one finds the ordering-dependent Tutte polynomial associated with the graph $G_1 \cup G_2$, performing a sequence of contractions and deletions as (1, 2, 3, 4, 5).

In a final comment, we want to address here the special case $\epsilon =\epsilon '=1$ and $\lambda _j = \lambda$ in view of possible interesting deformation of the ordinary Tutte polynomial. Consider the above example on $G_2$ and $G_1$, setting for simplicity $\beta =1$, and therefore we write

$$\begin{aligned} P^{1,1}(G_2;q;\lambda )= & {} q\Big [\alpha (\lambda ) \alpha (2\lambda ) + \alpha (2\lambda ) + \alpha (\lambda ) + q \Big ] \nonumber \\= & {} \sum _{B \subset G_2} q^{k(B)} \prod _{e\in B}\alpha (\lambda ^B_e), \end{aligned}$$

(B.20)

where $\lambda _e^B = c^B_e \lambda$ and $c_e^B$ is a positive integer. Observe that, for this example, $c_e^B$ becomes independent of B and so we reduce to the multivariate version of the Tutte polynomial [1] (to be explicit, we can redefine the edge labelling such that the polynomial $P^{1,1}$ coincides with multivariate Tutte polynomial). This is generally true when the adjacency matrix of the line graph is invariant under permutation of the edges. The case of clique graphs is a particular instance for which this is realized. Thus, for this type of graphs, we naturally loose the dependence on the order in which we perform the sequence of contraction deletion. We can check if this is also the case for the graph $G_1$ and find,

$$\begin{aligned} P^{1,1}(G_1;q;\lambda )= & {} q^{2} \alpha (\lambda ) \alpha (2\lambda ) \alpha (4\lambda ) + q \alpha (2\lambda ) \alpha (4\lambda ) + q \alpha (\lambda ) \alpha (2\lambda ) \nonumber \\&+ q \alpha (\lambda ) \alpha (4\lambda ) + q^2 \alpha (4\lambda ) + q^2 \alpha (\lambda ) + q^2 \alpha (2\lambda ) + q^3 \nonumber \\= & {} \sum _{B \subset G_1} q^{k(B)} \prod _{e\in B}\alpha (\lambda _e), \end{aligned}$$

(B.21)

confirming our expectations. In conclusion, in this particular limit, and for particular graphs, the order-independence of the generalized Tutte polynomial can be effectively restored.