I’m going to start off with two big caveats:

- This is not Duursma‘s definition, it’s mine.
- I’m not convinced (yet?) that it’s a useful idea to examine such a zeta function.

So that’s your warning – you may be wasting your time reading this!

The Duursma zeta function of a linear block (error-correcting) *code* is due to Iwan Duursma and is a fascinating object of study. (More precisely, it’s defined for “formal” linear block codes, ie, defined via a weight enumerator polynomial with a suitable MacWilliams-like functional equation.) Sometimes it satisfies an analog of the Riemann hypothesis and sometimes it doesn’t. And sometimes that analog is still an open question.

**Duursma zeta function of a code**

Before we define the Duursma zeta function of a graph, we introduce the Duursma zeta function of a code.

Let be an code, ie a linear code over of length , dimension , and minimum distance . In general, if is an -code then we use for the parameters of the dual code, . It is a consequence of Singleton’s bound that , with equality when is an MDS code. Motivated by analogies with local class field theory, in [Du] Iwan Duursma introduced the (Duursma) *zeta function* :

where is a polynomial of degree , called the *zeta polynomial* of . We next sketch the definition of the zeta polynomial.

If denotes the dual code of , with parameters then the *MacWilliams identity* relates the weight enumerator of to the weight enumerator of :

A polynomial for which

is a (Duursma) zeta polynomial of .

**Theorem** (Duursma): If be an code with and then the Duursma zeta polynomial exists and is unique.

See the papers of Duursma for interesting properties and conjectures.

**Duursma zeta function of a graph**

Let be a graph having vertices and edges. We define the zeta function of via the Duursma zeta function of the binary linear code defined by the cycle space of .

**Theorem** (see [DKR], [HB], [JV]): The binary code generated by the rows of the incidence matrix of is the cocycle space of over , and the dual code is the cycle space of :

Moreover,

(a) the length of is , dimension of is , and the minimum distance of is the edge-connectivity of ,

(b) length of is , dimension of is , and the minimum distance of is the girth of .

Call the *cycle code* and the *cocycle code*.

Finally, we can introduce the (Duursma) *zeta function* :

where is the Duursma polynomial of .

**Example**: Using SageMath, when , the wheel graph on 5 vertices, we have

All its zeros are of absolute value .

**References**

[Du] I. Duursma, *Combinatorics of the two-variable zeta function*, in **Finite fields and applications**, 109–136, Lecture Notes in Comput. Sci., 2948, Springer, Berlin, 2004.

[DKR] P. Dankelmann, J. Key, B. Rodrigues, *Codes from incidence matrices of graphs*, Designs, Codes and Cryptography 68 (2013) 373-393.

[HB] S. Hakimi and J. Bredeson, *Graph theoretic error-correcting codes*, IEEE Trans. Info. Theory 14(1968)584-591.

[JV] D. Jungnickel and S. Vanstone, *Graphical codes revisited*, IEEE Trans. Info. Theory 43(1997)136-146.

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