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Assume we have a graph $G=(V,E)$.

The Ihara zeta function $Z(G,u)$ is of form $$\frac1{\displaystyle\sum_{i=0}^{2|E|}c_iu^i}$$

A graph which has $|E|$ edges cannot have a simple cycle of length bigger than $|E|$.

So what do the coefficients $c_i$ mean for $i>|E|$?

In particular as an example what does $c_{2n}=1$ mean for an $n$-cycle graph?

Sorry I should have elaborated more.

I know of the formula $$ c_{2m}=(-1)^{m-n}\prod_{v_i \in V}(\deg(v_i)-1)$$ given in a PhD thesis. I am looking for more down to earth cycle interpretation of $c_{2m}$ and $c_{i}$ for $i>|E|$ in general.

I am aware of the document http://msp.org/involve/2008/1-2/involve-v1-n2-p08-p.pdf I just do not understand the interpretation of $c_{2m}$ for cycle graphs there since no general interpretation is given for $c_{i}$ for $i>|E|$ in general. Only interpretation seems to be twice sum of disjoint cycles or four times sum of overlapping cycles whose sum is $2m$. However twice or four times an integral value are greater than $1$ which is the value of $c_{2m}$ for cycle graphs.

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  • $\begingroup$ Do you mean $i > |V|$?. There can't be simple cycle of order greater than $|V|$. $\endgroup$
    – joro
    Commented Apr 12, 2016 at 5:15
  • $\begingroup$ I doubt there can be anything simpler for $C_n$ than Example 18 from the paper. $\endgroup$
    – joro
    Commented Apr 12, 2016 at 8:03
  • $\begingroup$ @joro I am trying to understand what they say in a more general framework. What does $C_i$ mean for higher $i$ in more down to earth terms in terms of cycles? do you understand my twice and four times confusion? This is the paper's standard interpretation. all these are bigger than $1$. $\endgroup$
    – Turbo
    Commented Apr 12, 2016 at 8:11
  • $\begingroup$ $C_n$ means the cycle graph on $n$ vertices. $\endgroup$
    – joro
    Commented Apr 12, 2016 at 8:22
  • $\begingroup$ Ok typo I meant coefficient $c_{2n}$ for cycle graph $C_n$. $\endgroup$
    – Turbo
    Commented Apr 12, 2016 at 8:50

1 Answer 1

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The paper The coefficients of the Ihara zeta function answers your question about the leading coefficient.

Let $n=|V|,m=|E|$. For cycles, we have $n=m$.

On p. 6 of the pdf: $$ c_{2m}=(-1)^{m-n}\prod_{v_i \in V}(\deg(v_i)-1)$$

Example 18 on p 12. of the pdf discusses $C_n$.

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