In spectral graph theory, there are many applications related to primes. In particular for Ramanujan graph Lubotzky, Phillips and Sarnak showed how to construct an infinite family of (p+1)-regular Ramanujan graphs, whenever p is a prime number congruent to 1 modulo 4. For more details see the below link:

Article Ramanujan Graphs

The famous Lucas’ theorem concerning congruences of binomial coefficients modulo primes and its generalizations play an important role in Combinatorics, and in particular in Combinatorial Number Theory. For more information, see my survey article: Romeo Meštrović, Lucas’ theorem, its generalizations, extensions and applications (1878-2014); available at https://arxiv.org/pdf/1409.3820.pdf

(preprint arXiv, September 2014, 51 pages); also available at https://www.researchgate.net/publication/265644350_Lucas'_theorem_its_generalizations_extensions_and_applications_1878--2014).

Wolstenholme’s theorem plays a fundamental role in Combinatorial Number Theory. A survey of Wolstenholme’s type congruences (modulo primes and prime powers), related problems and conjectures are presented in my article: Romeo Meštrović, Wolstenholme’s theorem: its generalizations and extensions in the last hundred and fifty years (1862-2012); available at https://arxiv.org/pdf/1111.3057.pdf (preprint arXiv, December 2011, 31 pages; also available at https://www.researchgate.net/publication/51967834_Wolstenholme's_theorem_Its_Generalizations_and_Extensions_in_the_lasthundred_and_fifty_years_1862--2012.

There are some connections between number theory and graph theory specially in the topic of prime labeling.