TUYMAADA 2023 J5

A graph contains $p$ vertices numbered from $1$ to $p$ , and $q$ edges numbered from $p + 1$ to $p + q$ . It turned out that for each edge the sum of the numbers of its ends and of the edge itself equals the same number $s$ . It is also known that the numbers of edges starting in all vertices are equal. Prove that $$s = \dfrac{1}{2} (4p+q+3).$$