Further results on square sum graph

A $(p,q)$ graph $G$ is said to be square sum, if there exists a bijection $f: V(G)rightarrow{0,1,2,dots,p-1}$ such that the induced function $ f^*: E(G)rightarrow N $ defined by $f^*(uv)= (f(u))^2+(f(v))^2,$ for every $uvin E(G)$ is injective. In this paper we establish that if $G$ is a square sum graph then $Gcup P_m$ is square sum, $(K_{m,n})^2$ is square sum if and only if $m+nleq 5$ and $W_n^2$ is square sum if and only if $nleq 5.$ We proved that shadow graph and split graph of $P_n$ and $K_{1,n}$ are square sum for every $nin N.$ Also union of paths, the sequential join of some classes of square sum graph, $mK_{1,n}$ for $m,nin N$ and $P_nodot K_2$(attaching $K_2$ to each vertex of $P_n$) are some classes of square sum graphs.