What is the Möbius Function for graphs?

Question

About 4 minutes into his video on chordal graphs, Donald Knuth mentions a variant of the Möbius Function for graphs. I was curious what it meant outside the context of number theory?

Answer 1

The Möbius function in number theory, though older, is actually a special case of a more general notion of Möbius function. If $\langle P,\preceq\rangle$ is a poset, a closed interval of $P$ is a set of the form

$$[a,b]=\{x\in P:a\preceq x\preceq b\}\;;$$

$P$ is locally finite if each of its closed intervals is finite. Let $\mathscr{I}$ be the set of non-empty closed intervals of $P$, and fix a commutative ring $R$ with unity, the ring of scalars; $\mathscr{I}$ and $R$ together give rise to an incidence algebra $A$ whose elements are the functions $f:\mathscr{I}\to R$. For $f,g\in A$ and $r\in R$ the sum $f+g$ and scalar multiple $rf$ are defined pointwise in the obvious way; a product is defined by

$$(f\mathrel{*}g)\big([a,b]\big)=\sum_{a\preceq x\preceq b}f\big([a,x]\big)g\big([x,b]\big)\;.$$

$A$ has a multiplicative identity $\delta$:

$$\delta\big([a,b]\big)=\begin{cases} 1,&\text{if }a=b\\ 0,&\text{otherwise}\;. \end{cases}$$

Finally, $A$ has a Möbius function $\mu$ that can be defined recursively as follows:

$$\mu\big([a,b]\big)=\begin{cases} 1,&\text{if }a=b\\ -\sum\limits_{a\preceq x\prec b}\mu\big([a,x]\big),&\text{if }a\prec b\\ 0,&\text{otherwise}\;. \end{cases}$$

If $P=\Bbb Z^+$ and $\preceq$ is $\mid$ (‘divides’), $\mu\big([m,n]\big)$ turns out to be the number-theoretic $\mu\left(\frac{n}m\right)$. You’ll find other examples at the link given above. This PDF gives more details but is still quite elementary. Richard P. Stanley’s notes ‘An Introduction to Hyperplane Arrangements’, which can be found here, also introduce the general Möbius function as part of some very general machinery, one application of which is to chordal graphs (under the name supersolvable graphs).

Answer 2

There is a Moebius function defined on any partially ordered set. The number theory one is the function associated to the set formed by the non-negative integers, partially ordered by divisibility. For details from this view point see, .e.g., http://en.wikipedia.org/wiki/Incidence_algebra or http://www.sfu.ca/~mdevos/notes/comb_struct/mobius.pdf

I don't have time to go through Knuth's lecture, so I cannot be sure what partial order he is using.