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Given the Cayley graph of a finitely generated group $G$, with respect to apresentation $G^{\alpha}$ with $n$ generators, the quotient of the rank of thefundamental group of subgraphs of the Cayley graph by the cardinality of theset of vertices of the subgraphs gives rise to the definition of the normalizedcyclomatic quotient $\Xi (G^{\alpha})$. The asymptotic behavior of thisquotient is similar to the asymptotic behavior of the quotient of thecardinality of the boundary of the subgraph by the cardinality of the subgraph.Using Følner's criterion for amenability one gets that $\Xi (G^{\alpha})$vanishes for infinite groups if and only if they are amenable. When $G$ isfinite then $\Xi (G^{\alpha})=1/|G|$, where $|G|$'> is the cardinality of $G$,and when $G$ is non-amenable then $1-n\leq\Xi (G^{\alpha})\le 0$, with $\Xi(G^{\alpha})=1-n$ if and only if $G$ is free of rank $n$. Thus we see that onspecial cases $\Xi (G^{\alpha})$ takes the values of the Euler characteristicof $G$. Most of the paper is concerned with formulae for the value of $\Xi(G^{\alpha})$ with respect to that of subgroups and factor groups, and withrespect to the decomposition of the group into direct product and free product.Some of the formulae and bounds we get for $\Xi (G^{\alpha})$ are similar tothose given for the spectral radius of symmetric random walks on the graph of$G^{\alpha}$, but this is not always the case. In the last section of the paperwe define and touch very briefly the balanced cyclomatic quotient, which isdefined on concentric balls in the graph and is related to the growth of $G$.

The normalized cyclomatic quotient associated with presentations of finitely generated groups