  An adjacency-crossing graph is a graph that can be drawn such that every two
edges that cross the same edge share a common endpoint. We show that the number
of edges in an $n$-vertex adjacency-crossing graph is at most $5n-10$. If we
require the edges to be drawn as straight-line segments, then this upper bound
becomes $5n-11$. Both of these bounds are tight. The former result also follows
from a very recent and independent work of Cheong et al.\cite{cheong2023weakly}
who showed that the maximum size of weakly and strongly fan-planar graphs
coincide. By combining this result with the bound of Kaufmann and
Ueckerdt\cite{KU22} on the size of strongly fan-planar graphs and results of
Brandenburg\cite{Br20} by which the maximum size of adjacency-crossing graphs
equals the maximum size of fan-crossing graphs which in turn equals the maximum
size of weakly fan-planar graphs, one obtains the same bound on the size of
adjacency-crossing graphs. However, the proof presented here is different,
simpler and direct.
