Infinite families of connected graphs with equal spectral radius

Abstract

We present three general mechanisms for constructing infinite families of connected graphs with equal spectral radius. The first method generates families of uniformly loaded cycles by coalescing copies of a vertex rooted connected graph at uniformly spaced vertices of a cycle. The second method is similar to the first one, except that now the load is an edge-rooted connected graph and the operation of vertex-coalescence is changed to edge-coalescence. The third method generates families of bracelets. A bracelet resembles somehow a loaded cycle but the construction mechanism is different in spirit. Among many other results, we show that if a real number r is the spectral radius of a connected graph that is not a tree, then r is realized as spectral radius by infinitely many connected graphs.