The maximum number of edges in a graph with matching number $m$ and maximum degree $d$ has been determined in Chvátal and Hanson (1976) and Balachandran and Khare (2009), where some extremal graphs have also been provided. Then, a new question has emerged: how the maximum edge count is affected by forbidding some subgraphs occurring in these extremal graphs? In Ahanjideh et al. (2022), the problem is solved in triangle-free graphs for $d\ge m$, and for $d<m$ with either $Z\left(d\right)\le m<2d$ or $d\le 6$, where $Z\left(d\right)$ is approximately $5d/4$. The authors derived structural properties of triangle-free extremal graphs, which allows us to focus on constructing small extremal components to form an extremal graph. Based on these findings, in this paper, we develop an integer programming formulation for constructing extremal graphs. Since our formulation is highly symmetric, we use our own implementation of Orbital Branching to reduce symmetry. We also implement our integer programming formulation so that the feasible region is restricted iteratively. Using a combination of the two approaches, we expand the solution into $d\le 10$ instead of $d\le 6$ for $m>d$. Our results endorse the formula for the number of edges in all extremal triangle-free graphs conjectured in Ahanjideh et al. (2022).

图中具有匹配数的最大边数$米$和最大程度$d$Chvátal 和 Hanson (1976) 以及 Balachandran 和 Khare (2009) 已确定，其中还提供了一些极值图。那么，一个新的问题出现了：禁止这些极值图中出现某些子图对最大边数有何影响？在 Ahanjideh 等人中。(2022)，问题在无三角形图中得到解决$d\ge 米$，并且对于$d<米$与任一$Z（d）\le 米<2d$或者$d\le 6$， 在哪里$Z（d）$大约是$5d/4$。作者推导了无三角形极值图的结构特性，这使我们能够专注于构造小的极值分量以形成极值图。基于这些发现，在本文中，我们开发了一种用于构造极值图的整数规划公式。由于我们的公式是高度对称的，因此我们使用自己的轨道分支实现来减少对称性。我们还实现了整数规划公式，以便迭代地限制可行区域。结合使用这两种方法，我们将解决方案扩展为$d\le 10$代替$d\le 6$为了$米>d$。我们的结果支持 Ahanjideh 等人推测的所有极值无三角形图中的边数公式。（2022）。