In this paper, we consider the implications of spectrum heterogeneity on connectivity and routing in a Cognitive Radio Ad Hoc Network (CRAHN). We study the Laplacian spectrum of the CRAHN graph when the activity of primary users is considered. We introduce the cognitive algebraic connectivity, i.e., the second smallest eigenvalue of the Laplacian of a graph, in a cognitive scenario. Throughout this notion we provide a methodology to evaluate the connectivity of CRAHNs and consequently introduce a utility function that is shown to be effective in capturing key characteristics of CRAHN paths. This model provides a unique metric that captures network connectivity, path length, and impact of primary users. Moreover, the proposed metric penalizes paths where spectrum band switchings are highly probable. We design all the components of our routing framework, named Gymkhana, and we present a twofold performance verification: one from a topological perspective to show all the potentialities of the proposed routing approach, and the other considering network traffic to evaluate the performance in terms of end-to-end delay and packet delivery ratio.
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