The minimal order of a graph which is not an induced subgraph of the given graph. For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices. By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number [[St000097]].

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k$$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.

The Frankl number of a lattice. For a lattice $L$ on at least two elements, this is $$\max_x(|L|-2|[x, 1]|),$$ where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.

The global dimension of the incidence algebra of the lattice over the rational numbers.

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.