The famous Goldbach conjecture remains open for nearly three centuries. Recently Goldbach graphs are introduced to relate the problem with the literature of Graph Theory. It is shown that the connectedness of the graphs is equivalent to the affirmative answer of the conjecture. Some modified version of the graphs, say, near Goldbach graphs are shown to be Hamiltonian for small number of vertices. In this context, we introduce a class of graphs, namely, prime multiple missing graphs such that near Goldbach graphs are finite intersections of these graphs. We study these graphs for primes 3,5 and in general for any odd prime p. We prove that these graphs are connected with diameter at most 3 and Hamiltonian for even (>2) vertices. Next the intersection of prime multiple missing graphs for primes 3 and 5 are studied. We prove that these graphs are connected with diameter at most 4 and they are also Hamiltonian for even (>2) vertices. We observe that the diameters of finite Goldbach graphs and near Goldbach graphs are bounded by 5 (up to 10000 vertices). We believe further study on these graphs with big data analysis will help to understand structures of near Goldbach graphs.

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