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An eigenvalue of a graph $G$ is called a main eigenvalue if it has aneigenvector the sum of whose entries is not equal to zero. It is well knownthat a graph $G$ has exactly two main eigenvalues if and only if there exists aunique pair of integers $a$ and $b$ such that $\sum_{u\in N(v)}d(u)=ad(v)+b$for every vertex $v\in V(G)$. We collect such connected graph $G$ in the set$\mathscr{G}(a,b)$. In this paper, we mainly focus to the existence of such $a$and $b$, and give the necessary and sufficient condition for$\mathscr{G}(a,b)\neq\emptyset$. In addition, we give the bound for the vertexdegrees of $G\in\mathscr{G}(a,b)$ and use the bound to characterize the graphsin $\mathscr{G}(a,b)$ for some feasible pairs $(a,b)$.

The existence of the graphs that have exactly two main eigenvalues