In this paper, we introduce the first learning-free model reuse task within the non-Euclidean domain, termed as Deep Graph Mating (Grama). We strive to create a child Graph Neural Network (GNN) that integrates knowledge from pre-trained parent models without requiring re-training, fine-tuning, or annotated labels. To this end, we begin by investigating the permutation invariance property of GNNs, which leads us to develop two vanilla approaches for Grama: Vanilla Parameter Interpolation (VPI) and Vanilla Alignment Prior to Interpolation (VAPI), both employing topology-independent interpolation in the parameter space. However, neither approach has achieved the anticipated results. Through theoretical analysis of VPI and VAPI, we identify critical challenges unique to Grama, including increased sensitivity to parameter misalignment and further the inherent topology-dependent complexities. Motivated by these findings, we propose the Dual-Message Coordination and Calibration (DuMCC) methodology, comprising the Parent Message Coordination (PMC) scheme to optimise the permutation matrices for parameter interpolation by coordinating aggregated messages, and the Child Message Calibration (CMC) scheme to mitigate over-smoothing identified in PMC by calibrating the message statistics within child GNNs. Experiments across diverse domains, including node and graph property prediction, 3D object recognition, and large-scale semantic parsing, demonstrate that the proposed DuMCC effectively enables training-free knowledge transfer, yielding results on par with those of pre-trained models.