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A generalized strongly regular graph of grade p, as a generalization of strongly regular graphs, is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values. In this paper, we study generalized strongly regular graphs of grade 2 and provide some inequalities for the eigenvalues of them. In particular, we investigate a special family of generalized strongly regular graphs of grade 2, i.e., semi-strongly regular graphs. We obtain a relation between the parameters and two inequalities for the eigenvalues of these graphs. We also present some constructions of generalized strongly regular graphs based on Cayley graphs, graph operations and association schemes, respectively. [ABSTRACT FROM AUTHOR]

Let Meta( H > T, λ) denote the set of all integers v such that there exists a ( H > T)- GM ( v). In this paper, the set Meta( H > T, λ) will be completely determined for the following 21 pairs ( H, T) = ( H, P), ( H, 2 P), ( H, P) and ( H, P), where $${P_2 \subset H_1 \subseteq K_4, H_2 \in \{K_4, C_4, K_3+e, P_4\} = {\mathcal {H}}, H_3 \in {\mathcal {H}} \cup \{K_3, K_{1,3} \}}$$ and $${H_4 \in \{K_4, C_4\}}$$ . [ABSTRACT FROM AUTHOR]