Graphs with Small Independence Number Minimizing the spectral radius

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications 431 (2009) 937–945] determined the connected graphs of order n with independence number α ∈ {1, 2, n 2 , n 2 + 1, n − 3, n − 2, n − 1} which minimize the spectral radius. In this paper, we show that the graph obtained from a path of order α by blowing up each vertex to a clique of order k minimizes the spectral radius among all connected graphs of order kα with independence number α for α = 3, 4 and conjecture that this is true for all α ∈ N.