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 vertexrooted 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.