Vertex-weighted $(k_{1},k_{2})$ $E$-torsion Graph of Quasi Self-dual Codes

Abstract

In this paper, we have introduced a graph $G_{EC}$ generated by type-$(k_{1},k_{2})$ $E$-codes which is $(k_{1},k_{2})$ $E$-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 $k_{1}=0$ and $k_{2}=0$ and calculated the degrees of every vertex and the number of edges of $G_{EC}$. 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 $(k_{1},k_{2})$ $E$-torsion graph, where the weight of every vertex is the weight of the code word corresponding to the vertex.