The pebbling number of a connected graph.

The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. Such a partition always exists because of a construction due to Dudek and Pralat [1] and independently Pokrovskiy [2].

The coalition number of a graph. This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.

The degree of the graph. This is the maximal vertex degree of a graph.

The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.

The length of a longest path in a graph.

The harmonious chromatic number of a graph. A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.

The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. Put differently, for every vertex $v$ of such a path $P$, there is a vertex $w\in P$ and a vertex $u\not\in P$ such that $(v, u)$ and $(u, w)$ are edges. The length of such a path is $0$ if the graph is a forest. It is maximal, if and only if the graph is obtained from a graph $H$ with a Hamiltonian path by joining a new vertex to each of the vertices of $H$.