EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 5, 2020, 1097-1109 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday On Tosha-degree of an edge in a graph R. Rajendra 1, P. Siva Kota Reddy 2,∗ 1 Department of Mathematics, Mangalore University, Mangalagangothri, Karnataka, India 2 Department of Mathematics, Sri Jayachamarajendra College of Engineering, JSS Science and Technology University, Mysuru, Karnataka, India 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. 2020 Mathematics Subject Classifications: 05Cxx, 05C07, 05C50. Key Words and Phrases: Adjacency matrix, degree of a vertex, energy, line graph, Tosha-degree of an edge 1. 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 de- gree of a vertex v ∈ V (G), denoted by d(v) or dG(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 ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3710 Email addresses: rrajendrar@gmail.com (R. Rajendra), pskreddy@jssstuniv.in (P. S. K. Reddy) https://www.ejpam.com 1097 c© 2020 EJPAM All rights reserved. R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1098 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 λ1, λ2, . . . , λn are the eigenvalues of A(G), the energy of G is defined as E(G) = n∑ i=1 |λi|. 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 re- sults. A signed graph is an ordered pair Σ = (G, σ), where G = (V,E) is a graph called the underlying graph of Σ and σ : E → {+,−} is a function. A marking of Σ is a function µ : V (G)→ {+,−}. In [4], we have also defined the Tosha-degree equivalence graph of a graph which is moti- vated 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 e1e2 in T (Σ), σ′(e1e2) = σ(e1)σ(e2). 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 kth iterated Tosha-degree equivalence signed graphs. Fur- ther, we have presented the structural characterization of Tosha-degree equivalence signed graphs. R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1099 2. 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 TG(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 T (α) = d(u) + d(v)− 2 (1) (ii) If α is a self-loop, then u = v and T (α) = d(u)− 2 (2) 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 2(δ(G)− 1) ≤ T (α) ≤ 2(∆(G)− 1); (b) if α is a self-loop, then δ(G)− 2 ≤ T (α) ≤ ∆(G)− 2. Corollary 1. [4] If G is a simple graph and α is an edge in G, then T (α) = dL(G)(α) (3) where dL(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). R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1100 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. u u u�� �� α β γ G Figure 1: Graph containing odd number of odd Tosha-degree edges. 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. Corollary 3. If α and β are parallel edges in a graph G, then T (α) = T (β) in G. Proof. The proof follows by Proposition 1. 2.1. T -line graph of a multigraph Definition 3. A multigraph is a graph in which multiple edges (parallel edges) are per- mitted 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 ad- jacent 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 3, it follows that, when two distinct edges α and β are k-adjacent in a multigraph G, we have, k = { 1, if α and β are not parallel; 2, if α and β are parallel. . R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1101 Definition 4. Given a multigraph G = (V,E), the T -line graph of G denoted by TL(G), is a graph with vertex set E; two distinct vertices α and β are k-adjacent in TL(G) if and only if their corresponding edges in G are k-adjacent. From the Definition 4, it is clear that, (a) TL(G) is also a multigraph, (b) if G is a simple graph, then TL(G) is nothing but L(G). Proposition 2. Let G be a multigraph and α be a vertex in TL(G) (so α is an edge in G). Then dTL(G)(α) = dG(u) + dG(v)− 2 = TG(α) (4) 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 TL(G) of G is even. Since the vertices in TL(G) are corresponding to the edges in G, by Eq.(4), the number of odd Tosha-degree edges in G is even. 3. 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 K2 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. Proposition 3. A simple connected graph G has a zero edge if and only if G ∼= K2. 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 d(u) + d(v)− 2 = 0 (5) Since G is connected, d(u) ≥ 1 and d(v) ≥ 1; from Eq.(5), d(u) = 1 and d(v) = 1. There- fore, there is no other edge in G incident to u and v. So G has only one edge α. Since G is connected, G ∼= K2. Conversely, if G ∼= K2, then clearly G is a simple connected graph having only one edge whose Tosha-degree is zero. R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1102 Corollary 5. A simple connected graph G with two or more edges, has no zero edge. Hence T (α) ≥ 1, ∀α ∈ E(G). Proof. Follows from Proposition 3. Corollary 6. A simple graph G has no zero edge if and only if either G 6∼= K2 or no component of G is isomorphic to K2 or no component of G is of only one vertex with a self-loop. Proof. Follows from Proposition 3. 4. 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), T (α) = d(u) + d(v)− 2. Since T (α) is odd, d(u) 6= 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 ver- tices. The following graph is an example for such graphs, in which the edges are indicated by respective Tosha-degrees. 5. Tosha-even graphs Definition 7. A graph G is said to be Tosha-even if all its edges are of even Tosha-degree. We recall the following proposition from [4]. Proposition 4. [4, Proposition 2.15] If G is an Euler graph, then all edges in G are of even Tosha-degree. R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1103 Corollary 8. Euler graphs are Tosha-even. 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, K4. 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 Tosha- even 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. 6. Tosha-adjacency matrix of a graph Definition 8. If G is a graph with n vertices v1, . . . , vn and no parallel edges. The Tosha- adjacency matrix of the graph G is an n × n matrix AT (G) = (tij) defined over the ring of integers such that tij = { T (vivj), if vivj ∈ E 0, otherwise. Observations: u u6 6 u @ @ @ @ @ @ @@ 5 7 7 7 G Figure 2: A degree colorable non-Tosha-regular graph with 3 vertices. u u u u6 6 3 5 3 16 6 6 2 G Figure 3: A degree colorable Tosha-even graph which is not an Euler graph. R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1104 (i) By the definition of the Tosha-degree of an edge, we have T (vivj) =  d(vi) + d(vj)− 2, if vivj ∈ E and i 6= j; d(vi)− 2, if vivj ∈ E and i = j; 0, if vivj /∈ E. Therefore, tij = tji. Therefore AT (G) is a real symmetric matrix. (ii) The entries along the principal diagonal of AT (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(AT (G)) = 0. In this case, if µ1, µ2, . . . , µn are the eigenvalues of AT (G), then n∑ i=1 µi = 0. (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 AT (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 AT (G) = l ·A(G). Proof. Suppose that G is l-Tosha-regular. Then T (α) = l, for all α ∈ E(G). Let A(G) = (aij) and AT (G) = (tij) be the adjacency matrix and the Tosha-adjacency matrix of G, respectively. Then by the definition of the Tosha-adjacency matrix AT (G), we have tij = { l, if vivj ∈ E 0, otherwise = l · aij . Therefore, AT (G) = l ·A(G). R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1105 Corollary 9. If a graph G with n vertices is r-regular, then AT (G) = 2(r − 1)A(G). 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, AT (G) = 2(r − 1)A(G). Corollary 10. A graph G is 1-Tosha-regular if and only if AT (G) = A(G). Proof. (⇐:) Suppose that for a graph G, AT (G) = A(G). Then by the definitions of AT (G) and A(G), it follows that, T (α) = 1, ∀α ∈ E(G). Hence, G is 1-Tosha-regular. (⇒:) Follows by Theorem 2. 7. Tosha-energy of a graph Definition 9. Let G be graph with n vertices v1, . . . , vn and no parallel edges. Let µ1, µ2, . . . , µn be the eigenvalues of the Tosha-adjacency matrix AT (G) of G. The Tosha- energy of G, denoted by ET (G), is defined as ET (G) = n∑ i=1 |µi|. (6) 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 ET (G) = l · E(G) (7) 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 AT (G) = l ·A(G) (8) where A(G) is the adjacency matrix of G. For brevity we write A for A(G) and AT for AT (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 AT . From Eq.(8) we have, det(AT − µI) = 0 ⇐⇒ det ( lA− µI ) = 0 ⇐⇒ ln det ( A− µ l I ) = 0 ⇐⇒ det ( A− µ l I ) = 0. R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1106 Therefore, µ is an eigenvalue of AT if and only if µ l is an eigenvalue of A. Let µ1, µ2, . . ., µn be the eigenvalues of the AT . Then µ1 l , µ2 l , . . ., µn l are the eigenvalues of A and the Tosha-energy of G is ET (G) = n∑ i=1 |µi| = l · n∑ i=1 ∣∣∣µi l ∣∣∣ = l · E(G). Case (ii): When l = 0. From Eq.(7), AT = 0 and so zero is the only eigenvalue of AT of multiplicity n. In this case, ET (G) = 0 = 0 · E(G). Corollary 11. The Tosha-energy of an r-regular graph G with n vertices is given by ET (G) = 2(r − 1)E(G) (9) 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. Corollary 12. (i) For the complete graph Kn on n > 1 vertices, ET (Kn) = 2(n− 2)E(Kn) = 4(n− 1)(n− 2). (ii) For the cycle graph Cn on n > 1 vertices, ET (Cn) = 2E(Cn) = 4 n−1∑ i=0 ∣∣∣∣cos ( 2πi n )∣∣∣∣ . (iii) For the complete bipartite graph Km,n, ET (Km,n) = (m+ n− 2)E(Km,n) = 2(m+ n− 2) √ mn. Proof. (i) The eigen values of A(Kn) are given below: eigen value → multiplicity → ( n− 1 −1 1 n− 1 ) Therefore E(Kn) = |n− 1|+ (n− 1)| − 1| = 2(n− 1). R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1107 Since Kn is an (n− 1)-regular graph, from Eq.(9) we have, ET (Kn) = 2(n− 2)E(Kn) = 2(n− 2) · 2(n− 1) = 4(n− 1)(n− 2). (ii) The eigen values of A(Cn) are 2 cos ( 2πi n ) , i = 0, 1, . . . , n− 1. Therefore E(Cn) = 2 n−1∑ i=0 ∣∣∣∣cos ( 2πi n )∣∣∣∣ . Since Cn is an 2-regular graph, from Eq.(9) we have, ET (Cn) = 2 · E(Cn) = 4 n−1∑ i=0 ∣∣∣∣cos ( 2πi n )∣∣∣∣ . (iii) The eigen values of A(Kn) are given below: eigen value → multiplicity → ( − √ mn 0 √ mn 1 n+m− 2 1 ) Therefore E(Km,n) = 2 √ mn. Since Km,n is an (m+ n− 2)-Tosha-regular graph, from Eq.(8) we have, ET (Km,n) = (m+ n− 2)E(Km,n) = 2(m+ n− 2) √ mn. Corollary 13. (i) For the path P2 of 2 vertices, ET (P2) = 0. (ii) For the path P3 of 3 vertices, ET (P3) = E(P3) = 2 √ 2. Proof. Since P2 = K1,1 and P3 = K2,1, (i) and (ii) follow immediately from Corol- lary 12 (iii). Theorem 3. Let G be a simple connected graph with at least one edge. Then AT (G) = A(G)⇐⇒ G = P3. Proof. (⇐:) If G = P3, 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, AT (G) = A(G). (⇒:) Suppose that AT (G) = A(G). Then G is 1-Tosha-regular and hence T (vivj) = 1, ∀ vivj ∈ E(G) R. Rajendra, P. S. K. Reddy / Eur. J. Pure Appl. Math, 13 (5) (2020), 1097-1109 1108 =⇒ d(vi) + d(vj)− 2 = 1, ∀ vivj ∈ E(G) =⇒ d(vi) = 3− d(vj), ∀ vivj ∈ E(G) Therefore, for any edge α in G with end vertices u and v, d(u) = 3− d(v) (10) 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. (11) 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. Let v be the other end vertex of α in G. Then by Eq.(10) and Eq.(11), either d(u) = 1 and d(v) = 2 or d(u) = 1 and d(v) = 2. If d(u) = 1 and d(v) = 2, there is another vertex w adjacent to v and d(w) = 1 (by above argument). There are no other vertices adjacent to the vertices u, v and w. So, G is a path with 3 vertices. A similar argument can be used for the case d(u) = 1 and d(v) = 2, to show that G is P3. 8. 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 e1, . . . , em. The edge-adjacency matrix of the graph G is an m×m matrix AE(G) = (xij) defined over the ring of integers such that xij = { k, if ei and ej are k−adjacent; 0, otherwise. Observations: (i) AE(G) is a {0, 1, 2}-matrix and it is real symmetric. If G is a simple graph, then AE(G) is a {0, 1}-matrix. (ii) The entries along the principal diagonal of AE(G) are all 0s. Therefore, tr(AE(G)) = 0. Hence if ν1, ν2, . . . , νm are the eigenvalues of AE(G), then m∑ i=1 νi = 0. (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 AE(G). REFERENCES 1109 Proposition 7. For a multigraph G, the edge-adjacency matrix of G is the adjacency matrix of the T -line graph of G. That is, AE(G) = A(TL(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, AE(G) = A(L(G)). Proof. For simple graph G, TL(G) = L(G) and so by Proposition 7 the result follows. Definition 12. Let G be graph with m edges e1, . . . , em. Let ν1, ν2, . . . , νm be the eigenval- ues of the edge-adjacency matrix AE(G) of G. The edge-energy of G, denoted by EE(G), is defined as EE(G) = m∑ i=1 |νi|. (12) Corollary 15. For a multigraph G, the edge-energy of G is the energy of the T -line graph of G. That is, EE(G) = E(TL(G)). Proof. Follows by Proposition 7 . Corollary 16. For a simple graph G, the edge-energy of G is the energy of the line graph of G. That is, EE(G) = E(L(G)). Proof. Follows by Corollary 14. Acknowledgements The authors would like to thank the referees for their invaluable comments and sug- gestions which led to the improvement of the manuscript. References [1] R B Bapat. Graphs and matrices. Springer, London, 2010. [2] F Harary. Graph theory. Addison-Wesley Pub. Co., Massachusetts, 1969. [3] R Rajendra and P Siva Kota Reddy. Tosha-degree equivalence signed graphs. Vladikavkaz. Mat. Zh., 22(2):48–52, 2020. [4] R Rajendra and P Siva Kota Reddy. Tosha-degree of an edge in a graph. Southeast Asian Bull. Math., accepted for publication.