On the Occasion of his 80th Birthday On Tosha-degree of an edge in a graph

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, edgeadjacency matrix and edge energy of a graph G and obtain some results. 2020 Mathematics Subject Classifications: 05Cxx, 05C07, 05C50.


Introduction
For standard terminology and notion in graphs and matrices, we refer the reader to the text-books of Harary [2] and Bapat [1]. The non-standard will be given in this paper as and when required.
Throughout this paper, G = (V, E) denotes a graph (finite and undirected) and V = V (G) and E = E(G) denote vertex set and edge set of G, respectively. The degree of a vertex v ∈ V (G), denoted by d(v) or d G (v), is the number of edges incident on v, with self-loops counted twice. A vertex of degree one is a pendant vertex and an edge incident onto a pendant vertex is a pendant edge. A graph G is r-regular if every vertex of G has degree r. The minimum degree δ(G) of a graph G is the minimum degree among all the vertices of G and the maximum degree ∆(G) of G is the maximum degree among all the vertices of G.
Two non-distinct edges in a graph are adjacent if they are incident on a common vertex. We consider that an edge in a graph is not adjacent to itself. The letters k, l, m, n, and r denote positive integers or zero.
The line graph L(G) of a simple graph with at least one edge is the graph (W, F ), where there is a one-to-one correspondence φ from E to W such that there is an edge between φ(α) and φ(β) if and only if the edges α and β are adjacent. We identify the set W by E.
The adjacency matrix of a graph G with n vertices is denoted by A(G). If A(G) is an n × n matrix and λ 1 , λ 2 , . . . , λ n are the eigenvalues of A(G), the energy of G is defined as 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. In this paper, 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.
In [4], we have also defined the Tosha-degree equivalence graph of a graph which is motivated us to extend this notion to signed graphs as follows: The Tosha-degree equivalence signed graph (See [3]) of a signed graph Σ = (G, σ) as a signed graph T (Σ) = (T (G), σ ), where T (G) is the underlying graph of T (Σ) is the Tosha-degree equivalence graph of G, where for any edge e 1 e 2 in T (Σ), σ (e 1 e 2 ) = σ(e 1 )σ(e 2 ). Hence, we shall call a given signed graph Σ as Tosha-degree equivalence signed graph if it is isomorphic to the Tosha-degree equivalence signed graph T (Σ ) of some sigraph Σ (See [3]). In [3], we offered a switching equivalence characterization of signed graphs that are switching equivalent to Tosha-degree equivalence signed graphs and k th iterated Tosha-degree equivalence signed graphs. Further, we have presented the structural characterization of Tosha-degree equivalence signed graphs.

Tosha-degree of an edge in a graph
In [4], R. Rajendra and P.S.K. Reddy have defined the Tosha-degree of an edge in a graph without multiple edges as follows: The Tosha-degree of an edge α in a graph G without multiple edges, denoted by T (α), is the number of edges adjacent to α in G, with self-loops counted twice. Here we allow graphs with multiple edges (multi-graphs) and the new definition of the Tosha-degree of an edge in a graph (with or without multiple edges) is given below: Definition 1. Let α be an edge in a graph G. The Tosha-degree of α, denoted by T (α) or T G (α), is the number of edges adjacent to α in G, where self-loops and edges parallel to α are counted twice.
By the Definition 1, for any edge α in a graph G, T (α) ≥ 0.

Definition 2.
A graph G is said to be a Tosha-regular graph if all edges are of equal Tosha-degree. We say that G is l-Tosha-regular, if T (α) = l, for all α ∈ E(G).
The following proposition is proved for graphs without parallel edges in [4]. This result is true for graphs having parallel edges also with respect to the Definition 1.

Proposition 1. [4]
Let α be an edge in a graph G with end vertices u and v.
(i) If α is not a self-loop, then Proof. The proof follows by the definition 1, and the definition of degree of a vertex.
Observation: By the Proposition 1, for an edge α in a graph G, it follows that, (a) if α is not a self-loop, then

Corollary 1. [4]
If G is a simple graph and α is an edge in G, then where d L(G) (α) is the degree of α as a vertex in the line graph L(G) of G.
Proof. Follows from the definition of L(G) and Eq.(1).

Corollary 2.
In a simple graph G, the number of odd Tosha-degree edges is even.
Proof. In any graph the number of odd degree vertices is even. So, the number of odd degree vertices in the line graph L(G) of G is even. Since the vertices in L(G) are corresponding to the edges in G, by Eq.(3) it follows that, the number of odd Tosha-degree edges in G is even. Remark 1. The Corollary 2 may not be true for the graphs having self-loops. There are graphs with odd number of edges and all edges are of odd Tosha-degree. For eg., consider the graph G given in Figure 1. The graph G has three edges namely, α, β and γ. We observe that T (α) = 1, T (β) = 3, T (γ) = 1 and hence all the edges in G are of odd Tosha-degree. Observation: Let α be an edge in a simple graph G. The addition of a parallel edge β to α gives a count plus two to the Tosha-degree of α and to the edges parallel to α, and a count plus one to non-parallel edges adjacent to α in the new graph G + β and Tosha-degrees of all other edges are unaltered G + β. Hence an odd (even) Thosha-degree edge γ remains odd (even) Tosha-degree in G + β, if it is not adjacent to α or γ = α in G. Proof. The proof follows by Proposition 1.

T -line graph of a multigraph
Definition 3. A multigraph is a graph in which multiple edges (parallel edges) are permitted between any pair of vertices. All multigraphs in this paper are loopless.
We say that two distinct edges α and β in a multigraph G are k-adjacent if they are adjacent and share k end vertices.
We say that two distinct vertices u and v in a multigraph G are r-adjacent if they are adjacent and the number of edges between them is r (i.e., r edges have common end vertices u and v).  From the Definition 4, it is clear that, Proposition 2. Let G be a multigraph and α be a vertex in T L(G) (so α is an edge in G). Then where u and v are end vertices of α in G.
Proof. Proof follows by the definitions 1, 3 and 4, and propositions 1 and 2.
Corollary 4. In a multigraph G, the number of odd Tosha-degree edges is even.
Proof. In any graph(multigraph) the number of odd degree vertices is even. So, the number of odd degree vertices in the line graph T L(G) of G is even. Since the vertices in T L(G) are corresponding to the edges in G, by Eq.(4), the number of odd Tosha-degree edges in G is even.

Zero edges in a graph
Definition 5. In a graph G, an edge α is said to be a zero edge if its Tosha degree is zero i.e., T (α) = 0.
Observations: The edge in the complete graph K 2 is a zero edge. The self-loop in the graph containing only one vertex and a self-loop attached to that vertex, is a zero edge. Proof. Suppose that G is a simple connected graph having a zero edge, say α = uv, where u and v are end vertices of α. Then Since G is connected, d(u) ≥ 1 and d(v) ≥ 1; from Eq.(5), d(u) = 1 and d(v) = 1. Therefore, there is no other edge in G incident to u and v. So G has only one edge α. Since G is connected, Conversely, if G ∼ = K 2 , then clearly G is a simple connected graph having only one edge whose Tosha-degree is zero.
Corollary 6. A simple graph G has no zero edge if and only if either G ∼ = K 2 or no component of G is isomorphic to K 2 or no component of G is of only one vertex with a self-loop.

Degree colorable graphs
In this section we consider self-loop free graphs (multigraphs).
Definition 6. A graph G is degree colorable if no two adjacent vertices have the same degree.
Theorem 1. If all the edges of a graph G are of odd Tosha-degree, then G is a degree colorable graph with even number of vertices.
Proof. Suppose that G is a graph in which all the edges are of odd Tosha-degree. By the corollaries 2 and 4, it follows that G has an even number of vertices. Let α be an edge in G with end vertices u and v. Then by Eq.(1) and Eq. (4), Since T (α) is odd, d(u) = d(v). Thus, no two adjacent vertices in G have the same degree. Therefore G is a degree colorable graph.
By Theorem 1, the following corollary is immediate.
Corollary 7. An l-Tosha-regular graph, where l is an odd positive integer, is degree colourable.

Remark 2.
There are degree colorable non-Tosha-regular graphs with odd number of vertices. The following graph is an example for such graphs, in which the edges are indicated by respective Tosha-degrees.

Tosha-even graphs
Definition 7. A graph G is said to be Tosha-even if all its edges are of even Tosha-degree.

Proof. Follows from the Proposition 4.
Remark 3. .The converse of the Corollary 8 is not true in general. There are connected graphs with even number of vertices and all vertices are of odd degree, for instance, K 4 . Such graphs are not Euler graphs, but are Tosha-even.
Proposition 5. There exist degree colorable Tosha-even graphs that are not Euler graphs.
Proof. The following graph G (see Figure 3) is an example of a degree colorable Toshaeven graph which is not an Euler graph. In G, the vertices and edges are indicated by their degrees and Tosha-degrees, respectively. We see that all vertices of G are of odd degree and hence G is not an Euler graph. But all edges are of Tosha-even, so G is a Tosha-even graph.

Tosha-adjacency matrix of a graph
Definition 8. If G is a graph with n vertices v 1 , . . . , v n and no parallel edges. The Toshaadjacency matrix of the graph G is an n × n matrix A T (G) = (t ij ) defined over the ring of integers such that Observations: (i) By the definition of the Tosha-degree of an edge, we have Therefore, t ij = t ji . Therefore A T (G) is a real symmetric matrix.
(ii) The entries along the principal diagonal of A T (G) are all 0s if and only if either G has no self-loops or G has only self loops that are zero edges. Hence if either G has no self-loops or G has only self loops that are zero edges, then tr(A T (G)) = 0. In this case, if µ 1 , µ 2 , . . . , µ n are the eigenvalues of A T (G), then (iii) If G has no zero edges, then the degree of a vertex equals the number of non-zero entries in the corresponding row or column; and the non-zero entry in the ij-th place gives the Tosha-degree of the corresponding edge incident to i-th and j-th vertices.
(iv) For a zero edge free graph G, the adjacency matrix A(G) can be obtained from the Tosha-adjacency matrix A T (G) by replacing all the non-zero entries by 1s. This is possible because, in a zero edge free graph Tosha-degrees of edges are non-zero. Thus, reconstriction of the graph from the Tosha-adjacency matrix is possible if the given graph has no zero edges.
Throughout this section G denotes a graph with no parallel edges.
Theorem 2. If a graph G with n vertices is l-Tosha-regular, then Proof. Suppose that G is l-Tosha-regular. Then T (α) = l, for all α ∈ E(G). Let A(G) = (a ij ) and A T (G) = (t ij ) be the adjacency matrix and the Tosha-adjacency matrix of G, respectively. Then by the definition of the Tosha-adjacency matrix A T (G), we have Therefore, A T (G) = l · A(G).
Corollary 9. If a graph G with n vertices is r-regular, then Proof. If a graph G with n vertices is r-regular, then G is 2(r − 1)-Tosha-regular(by [4, Corollary 2.6]) and hence by Theorem 2, A T (G) = 2(r − 1)A(G).

Tosha-energy of a graph
Definition 9. Let G be graph with n vertices v 1 , . . . , v n and no parallel edges. Let µ 1 , µ 2 , . . . , µ n be the eigenvalues of the Tosha-adjacency matrix A T (G) of G. The Toshaenergy of G, denoted by E T (G), is defined as Throughout this section G denotes a graph with no parallel edges.
Proposition 6. The Tosha-energy of an l-Tosha-regular graph G with n vertices is given by where E(G) is the energy of G.
Proof. Le G be an l-Tosha-regular graph with n vertices. Then by the Theorem 2, the Tosha-adjacency matrix of G is where A(G) is the adjacency matrix of G. For brevity we write A for A(G) and A T for A T (G). We consider two cases: (i) When l > 0 and (i) When l = 0. Case (i): When l > 0. Let µ be an eigenvalue of A T . From Eq.(8) we have, Therefore, µ is an eigenvalue of A T if and only if µ l is an eigenvalue of A. Let µ 1 , µ 2 , . . ., µ n be the eigenvalues of the A T . Then µ 1 l , µ 2 l , . . ., µ n l are the eigenvalues of A and the Tosha-energy of G is Case (ii): When l = 0. From Eq.(7), A T = 0 and so zero is the only eigenvalue of A T of multiplicity n. In this case, E T (G) = 0 = 0 · E(G).
Corollary 11. The Tosha-energy of an r-regular graph G with n vertices is given by where E(G) is the energy of G.
Proof. Let G be an r-regular graph with n vertices. By [4, Corollary 2.6] G is a 2(r − 1)-Tosha-regular graph. Then by Proposition 6, the proof follows.
(iii) The eigen values of A(K n ) are given below: Since K m,n is an (m + n − 2)-Tosha-regular graph, from Eq.(8) we have, Corollary 13. (i) For the path P 2 of 2 vertices, E T (P 2 ) = 0.
Theorem 3. Let G be a simple connected graph with at least one edge. Then Proof. (⇐:) If G = P 3 , then it has two edges and each of these are of Tosha-degree 1. Therefore, it is 1-Tosha-regular and hence by Theorem 2, A T (G) = A(G). (⇒:) Suppose that A T (G) = A(G). Then G is 1-Tosha-regular and hence Therefore, for any edge α in G with end vertices u and v, Since G is connected, d(v) > 0 and d(u) > 0, and from Eq.(10) we have, d(u) < 3; which implies d(u) = 1 or 2.
Let u be an arbitrary vertex in G. Since G is a simple connected graph with at least one edge, u is an end vertex of at least one edge say α.

Edge-adjacency matrix and edge-energy of a graph
Definition 10. We say that two distinct edges α and β in a graph G (where self-loops and parallel edges are allowed) are k-adjacent if they are adjacent and share k end vertices. We consider that an edge in a graph is not adjacent to itself.
Definition 11. If G is a graph with m edges e 1 , . . . , e m . The edge-adjacency matrix of the graph G is an m × m matrix A E (G) = (x ij ) defined over the ring of integers such that x ij = k, if e i and e j are k−adjacent; 0, otherwise. Observations: (i) A E (G) is a {0, 1, 2}-matrix and it is real symmetric. If G is a simple graph, then A E (G) is a {0, 1}-matrix. (iii) If G has no self-loops, then the Tosha-degree of an edge equals the sum of entries in the corresponding row or column of A E (G).
Proof. Follows by the definitions 4 and 11.
Corollary 14. For a simple graph G, the edge-adjacency matrix of G is the adjacency matrix of the line graph of G. That is, Proof. For simple graph G, T L(G) = L(G) and so by Proposition 7 the result follows.
Definition 12. Let G be graph with m edges e 1 , . . . , e m . Let ν 1 , ν 2 , . . . , ν m be the eigenvalues of the edge-adjacency matrix A E (G) of G. The edge-energy of G, denoted by E E (G), is defined as Corollary 15. For a multigraph G, the edge-energy of G is the energy of the T -line graph of G. That is, E E (G) = E(T L(G)).
Corollary 16. For a simple graph G, the edge-energy of G is the energy of the line graph of G. That is, E E (G) = E(L(G)).