Abstract
In the first half, we present a generalization of the graph theory prime-number theorem due to Audrey Terras et al. In the last half, we study relations between several densities of a subset consisting of primes in a graph, and by using the first result, we compute a distribution of a subset. This is an answer for the "research problem" in the page 197 of Terras' textbook "Zeta Functions of Graphs."
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