  Determining the maximum number of edges under degree and matching number
constraints have been solved for general graphs by Chv\'{a}tal and Hanson
(1976), and by Balachandran and Khare (2009). It follows from the structure of
those extremal graphs that deciding whether this maximum number decreases or
not when restricted to claw-free graphs, to $C_4$-free graphs or to
triangle-free graphs are separately interesting research questions. The first
two cases being already settled, respectively by Dibek, Ekim and Heggernes
(2017), and by Blair, Heggernes, Lima and D.Lokshtanov (2020). In this paper we
focus on triangle-free graphs. We show that unlike most cases for claw-free
graphs and $C_4$-free graphs, forbidding triangles from extremal graphs causes
a strict decrease in the number of edges and adds to the hardness of the
problem. We provide a formula giving the maximum number of edges in a
triangle-free graph with degree at most $d$ and matching number at most $m$ for
all cases where $d\geq m$, and for the cases where $d<m$ with either $d\leq 6$
or $Z(d)\leq m < 2d$ where $Z(d)$ is a function of $d$ which is roughly $5d/4$.
We also provide an integer programming formulation for the remaining cases and
as a result of further discussion on this formulation, we conjecture that our
formula giving the size of triangle-free extremal graphs is also valid for
these open cases.
