Quantitative measure of symmetry

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Let G be an undirected graph with big odd amount of nodes linked in a line. Denote central node as C. If we connect one extra node to node N, which is one edge from C, resulting graph will be more symmetric than graph with extra node, connected to node F, which is 10 edges from C. Is there area of mathematics which is associated with such quantitative measure of symmetry (not necessarily of graphs)?