The distance ideals of graphs are algebraic invariants that generalize the Smith normal form and the spectrum of several distance matrices associated with a graph. In general, distance ideals are not monotone under taking induced subgraphs. However, in [5] the characterizations of connected graphs with a trivial distance ideal over \(\mathbb {Z}[X]\) and over \(\mathbb {R}[X]\) were obtained in terms of induced subgraphs, where X is a set of variables associated with the vertices of the graph. Later, in [2], the first attempt was made to characterize the family of connected graphs with at most two trivial distance ideals over \(\mathbb {Z}[X]\). There, it was proven that these graphs are free of \(\textsf {odd-hole}_{7}\) graphs and a set \(\mathcal {F}\) of sixteen graphs, where \(\textsf {odd-hole}_{7}\) consists of all cycles of odd length of at least seven. Here, we give a characterization of the \(\lbrace \mathcal {F},\textsf {odd-hole}_{7}\rbrace\) -free graphs and prove that the \(\lbrace \mathcal {F},\textsf {odd-hole}_{7}\rbrace\) -free graphs are the graphs with at most two trivial distance ideals over \(\mathbb {Z}[X]\). As a byproduct, we also find that the determinant of the distance matrix of a connected bipartite graph is even.

Motivated by a problem asked by Richter and by the long standing Harary--Hill conjecture, we study the relation between the crossing number of a graph $G$ and the crossing number of its cone $CG$, the graph obtained from $G$ by adding a new vertex adjacent to all the vertices in $G$. Simple examples show that the difference $cr(CG)-cr(G)$ can be arbitrarily large for any fixed $k=cr(G)$. In this work, we are interested in finding the smallest possible difference; that is, for each nonnegative integer $k$, find the smallest $f(k)$ for which there exists a graph with crossing number at least $k$ and cone with crossing number $f(k)$. For small values of $k$, we give exact values of $f(k)$ when the problem is restricted to simple graphs and show that $f(k)=k+\Theta (\sqrt {k})$ when multiple edges are allowed.