Vertex-weighted (k₁,k₂) E-torsion Graph of Quasi Self-dual Codes

In this paper, we have introduced a graph GEC generated by type-(k₁,k₂) E-codes which is (k₁,k₂) E-torsion graph. The binary code words of the torsion code of C are the set of vertices, and the edges are defined using the construction of E-codes. Also, we characterized the graph obtained when k₁=0 and k₂=0 and calculated the degrees of every vertex and the number of edges of GEC. Moreover, we presented necessary and sufficient conditions for a vertex to be in the center of a graph given the property of the code word corresponding to the vertex. Finally, we represent every quasi-self dual codes of short length by defining the vertex-weighted (k₁,k₂) E-torsion graph, where the weight of every vertex is the weight of the code word corresponding to the vertex.