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The purpose of this work is to shed some new light on the spectral properties of the family of friendship graphs and, more generally, the family of weak-friendship graphs. A weak-friendship graph is a connected induced subgraph of a friendship graph. We characterize the spectral radius of a friendship graph and of each one of its connected induced subgraphs. We prove that the separability index of the family of weak-friendship graphs is equal to 2. We provide exact and asymptotic estimates for the number of complementarity eigenvalues of a weak-friendship graph.

Spectral radii of friendship graphs and their connected induced subgraphs