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Abstract
In an earlier paper, we have introduced the Tosha-degree of an edge in a graph without multiple edges and studied some properties. In this paper, we extend the definition of Tosha-degree of an edge in a graph in which multiple edges are allowed. Also, we introduce the concepts - zero edges in a graph, $T$-line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge-adjacency matrix and edge energy of a graph $G$ and obtain some results.
Highlights
For standard terminology and notion in graphs and matrices, we refer the reader to the text-books of Harary [2] and Bapat [1]
In our earlier paper [4], we have introduced the Tosha-degree of an edge in a graph without multiple edges, Rajendra-Reddy index of a graph and Tosha-degree equivalence graph of a graph, and studied some properties
The aim of this paper is to introduce the concepts: zero edges in a graph, T -line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge-adjacency matrix and edge energy of a graph G and obtain some results
Summary
For standard terminology and notion in graphs and matrices, we refer the reader to the text-books of Harary [2] and Bapat [1]. Two non-distinct edges in a graph are adjacent if they are incident on a common vertex. In our earlier paper [4], we have introduced the Tosha-degree of an edge in a graph without multiple edges, Rajendra-Reddy index of a graph and Tosha-degree equivalence graph of a graph, and studied some properties. We define Tosha-degree of an edge in a graph in which multiple edges are allowed. The aim of this paper is to introduce the concepts: zero edges in a graph, T -line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge-adjacency matrix and edge energy of a graph G and obtain some results. We have presented the structural characterization of Tosha-degree equivalence signed graphs
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