Connection graphs are a natural extension of Harary’s signed graphs. The Bakry-\’Emery curvature of the connected graph was introduced by Liu, M’unch, and Peyrimhoff to establish his Buser-type eigenvalue estimate of the connected Laplacian. In this paper, we reformulate the Bakry-\’Emery curvature of vertices. The connection graph for the smallest eigenvalues of a family of unitary equivalent curvature matrices. We further interpret this family of curvature matrices as matrix representations of newly defined curvature tensors with respect to different orthonormal bases of the tangent space at the vertices. , is a powerful extension of previous work by Cushing-Kamtue-Liu-Peyerimhoff and Siconolfi on curvature matrices of graphs. Furthermore, we study the Bakry-\’Emery curvature of the Cartesian product of connected graphs, reinforcing Liu’s previous results. Munch and Pelimhoff. Although the results of vertices with locally balanced structures cover previous studies, they reveal various interesting phenomena of locally unbalanced connected structures.
