Let $G$ be a vertex-transitive locally finite graph and $c\_n$ the number of self-avoiding walks in $G$ starting from some fixed vertex $v\_0$. One can easily see that $c\_{m+n} \leq c\_m c\_n$ and hence Fekete's subadditive lemma gives that there exists the limit $\mu := \lim\_{n\to\infty} c\_n^{1/n}$. This quantity is known in the literature as the **connective constant** of $G$.


My question is basically what is the source of this term? I.e. it seems to be a strange choice of words for this quantity, so is there some application where it plays a role in some kind of "connectedness" which would excuse the name?



Origin of the term "connective constant"
**Q:** Is there some application where $\mu$ plays a role in some kind of "connectedness" which would excuse the name?


**A:** The application is to crystalline structure. The name originates from Hammersley, who introduced [1,2] the "connective constant" to characterize the bonds in a crystal.



> 
> In a certain sense [the connective constant] measures the richness of
> the connexions in a crystal.
> 
> 
> 


For example, if each atom in a branching process has $M$ direct descendant atoms to each of which there is a one-way bond, then $\mu=\log M$.


[1] J. M. Hammersley, *Percolation processes II. The connective constant*, Proc. Camb. Phil. Soc. 53 (1957), 642–645.  

[2] S. R. Broadbent and J. M. Hammersley, [Percolation processes](https://courses.physics.ucsd.edu/2019/Spring/physics235/Broadbent_Hammersley.pdf) (1957).


