The extremal spectral radius of outerplanar graphs without Ck

An outerplanar graph is a graph which can be embedded in the plane so that all vertices are on one face, say the outer face. Let [Formula: see text] be a cycle of length [Formula: see text] with [Formula: see text]. Among the set of outerplanar graphs on [Formula: see text] vertices without [Formula: see text], we first obtain that the extremal graph having the maximum spectral radius contains a subgraph [Formula: see text] for [Formula: see text] and [Formula: see text] and then we deduce the unique extremal graph having the maximum spectral radius for [Formula: see text] and [Formula: see text], where [Formula: see text] is the star with [Formula: see text] vertices. Furthermore, for [Formula: see text], among the set of outerplanar graphs without [Formula: see text] on [Formula: see text] vertices, we get that the star [Formula: see text] is the unique extremal graph having the maximum spectral radius.