What is the maximal connected partial symmetry index of a connected graph of a given size?
Z. Janelidze, F. van Niekerk, J. Viljoen
TL;DR
This paper resolves the maximal connected partial symmetry index for connected graphs of a fixed size by introducing $CoPSI(G)$ as the count of all isomorphisms between connected induced subgraphs. It proves that among connected graphs with size $n$, the star $K_{1,n}$ maximizes $CoPSI(G)$, with equality only for $G \cong K_{1,n}$; it provides a closed-form formula $CoPSI(K_{1,n}) = 2n(n+1) + \sum_{i=0}^n {n \choose i}^2 i!$ and relates it to the partial symmetry index of the complete graph: $CoPSI(K_{1,n}) = 2n(n+1) + PSI(K_n)$. The paper also decomposes connected partial symmetries into singleton, edge, and third-type, showing that the third-type corresponds to non-singleton permutations of the edge-set, and discusses $CoPSI$ for paths and cycles. This establishes a new extremal characterization in graph symmetry and connects to integer sequences in OEIS.
Abstract
For a given graph, by its \emph{connected partial symmetry index} we mean the number of all isomorphisms between connected induced subgraphs of the graph. In this brief note we answer the question in the title.