Khattree introduced the concept of constant block sum partially balanced incomplete block designs (CBS-PBIBDs) in 2019. In this paper, we have tried to extend this concept to designs arising from different graph parameters. We have studied when some graph theoretical parameters such as diametral paths, maximum independent sets and geodetic sets taken as blocks yield CBS-PBIBD for some classes of graphs. We have also given the distance based association schemes for each design and found robustness of these designs using A-, D- and E-optimality criteria. Also we have disproved existence of a (v, β0, μ) design arising from Shrikhande graph proved by Walikar, Acharya and Shirkol by establishing a flaw in their proof and also attempted one of the questions posed by Khattree on CBS-PBIBD.