Summary: ``An $s$-bundle (where $s$ is a positive integer) is a connected graph, the vertex connectivity of which is at least $n - s$, where $n$ is the number of vertices in the graph. As a relaxation of the classical clique model, the $s$-bundle is relevant for representing cohesive groups with an emphasis on the connectivity of members; thus, it is of great practical importance. In this work, we investigate the fundamental problem of finding the maximum $s$-bundle from a given graph and present an effective branch-and-bound algorithm for solving this NP-hard problem. The proposed algorithm is distinguished owing to its new multi-branching rules, graph coloring-based bounding technique, and reduction rules using structural information. The experiments indicate that the algorithm outperforms the best-known approaches on a wide range of well-known benchmark graphs for different $s$ values. In particular, compared with the popular Russian Doll Search algorithm, the proposed algorithm almost doubles the success rate of solving large social networks in an hour when $s = 5$.''