The eigenvalues of the combinatorial Laplacian of graphs with boundaries and infinite graphs without boundary are studied. For a graph with boundary G =(V ∪∂V,E ∪∂E ), a sharp lower bound of the first eigenvalue λ₁(G ) is given provided G satisfies a general condition, the so called non-separation property. For an infinite graph G without boundary, the bottom of the spectrum, i.e., the infimum of the spectrum of the combinatorial Laplacian of G, denoted λ₀(G ), is estimated as

λ₀(G ) ≤ ¼ μ(G )²exp(μ(G )),

where μ(G ) is the exponential growth of G. As a corollary, if G is subexponential, λ₀(G )=0. On the contrary,λ₀(G ) >0 is shown for a simply connected infinite graph G with degree ≥4 at each vertex.