We study the fair allocation of graphical resources, where the resources are the vertices in a graph. Upon receiving a set of resources, an agent’s utility equals the weight of a maximum matching in the induced subgraph. We care about maximin share (MMS) fairness and envy-freeness up to one item (EF1). Regarding MMS fairness, the problem does not admit a finite approximation ratio for heterogeneous agents. For homogeneous agents, we design polynomial-time constant-approximation algorithms, and also note that significant amount of social welfare is sacrificed inevitably in order to ensure (approximate) MMS fairness. We then consider EF1 allocations whose existence is guaranteed. However, the social welfare guarantee of EF1 allocations cannot be better than 1/n for the general case, where n is the number of agents. Fortunately, for three special cases, two-agent, binary-weight and homogeneous-agent, we are able to design polynomial-time algorithms that also ensure a constant fraction of the maximum social welfare.