One generalization with a spectral graph theory flavor is **the Colin de Verdière Conjecture**, originating in

Colin de Verdière, Yves. "Sur un nouvel invariant des graphes et un critere de planarité." *Journal of Combinatorial Theory, Series B* 50, no. 1 (1990): 11-21. Journal link (English translation in this volume)

For a graph $G$ with $n$ vertices, consider an $n\times n$ symmetric matrix $M$ satisfying:

- For every $i\neq j$, $M_{ij} < 0$ if $\{i,j\}$ is an edge in $G$, and $M_{ij} = 0$ otherwise.

(This looks a bit like a generalization of the usual graph Laplacian matrix.)

The smallest eigenvalue of $M$ must (by Perron-Frobenius) have multiplicity equal to the number of connected components of $G$. So what about the multiplicity of the *second smallest* eigenvalue? Now this may depend on the choice of $M$, so let's consider its maximum over all such matrices $M$.

Since the diagonal of $M$ is unconstrained, we may assume that this second-smallest eigenvalue is $0$. Then we're asking about the largest possible corank of such a matrix $M$.

Finally, require of $M$ a sort of nondegeneracy condition, called the Strong Arnold Property:

- Within the space of $n\times n$ real symmetric matrices, the submanifold comprising those that satisfy the bulleted condition above and that comprising those with the same rank as $M$ intersect transversally at $M$.

(A theorem of van der Holst, Lovász and Schrijver gives an equivalent algebraic condition: The only symmetric matrix $X$ with $MX=0$ that is zero on the diagonal and on the edges of $G$ is $X=0$.)

The largest corank of a matrix $M$ satisfying both of the bulleted conditions above is the Colin de Verdière number of $G$, denoted $\mu(G)$. This parameter has some nice properties, e.g., it is monotonic with respect to graph minors.

Most remarkably, Colin de Verdière showed that $\mu(G) \le 3$ if and only if $G$ is planar (and that $\mu(G) \le 2$ iff $G$ is outerplanar) and put forward

**The Colin de Verdière Conjecture:** $\chi(G) \le \mu(G)+1$

Currently, the conjecture is known to hold for $\mu(G) \le 4$. This relies on current proofs of the 4-color theorem, of course, although a direct proof of the conjecture could conceivably offer a very different route to that result.

**ADDED:** Colin de Verdière showed $\mu(G)=n-1$ iff $G=K_n$. (This seems obvious, but does require checking the Strong Arnold Property.) Together with the minor-monotonicity mentioned above, this shows that the conjecture would follow as a special case of the Hadwiger Conjecture, as pointed out here in a comment by Gil Kalai. This is also noted by Colin de Verdière himself upon stating the conjecture!