This thesis consists of four parts.In the first part,we give two characterizations of the（p×q）-grid graphs as a co-edge-regular graphs with four distinct eigenvalues.To give the characterization of the（p × q）-grid graphs we introduce a new class of co-edge-regular graphs that gives a combinatorial generalization of co-edge-regular graphs with exactly four distinct eigenvalues.In the second part,we show the non-existence of a strongly regular graph with parameters（1911,270,105,27）,which is the largest open case of a set of feasible parameters of a strongly regular graphs with smallest eigenvalue-3.Neumaier showed that for any connected strongly regular graph with parameters（n,k,λ,μ）and smallest eigenvalue-θ,we have either μ≤θ3（2θ-3）or the graph is a complete multipartite,for an integer θ≥2.In the third part,we give a generalization of Neumaier result for a new class of regular graphs called sesqui-regular graphs.We show that for a connected sesqui-regular graph with parameters μ)and smallest eigenvalue at least-8,if k is large,then μ≤θ2（θ-1）or n-k-1≤（θ-1）2/4+1 holds.We use the theory of Hoffman graphs as a tool in this part.Koolen,Yang and Yang used the notion of s-integrability of graphs.They proved that there exists an integer κ＞0,such that if G is connected graph with minimal valency greater than or equal to κ and smallest eigenvalue at least-3,then G is 2-integrable.In the last part,we continue the work of Koolen,Yang and Yang,and we study the integrability of sesqui-regular graphs with smallest eigenvalue at least-3.