EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 773-783 ISSN 1307-5543 – ejpam.com Published by New York Business Global Relations Between Vertex–Edge Degree Based Topological Indices and Mve-Polynomial of r−Regular Simple Graph Kavi B. Rasool1,∗, Payman A. Rashed2, Ahmed M. Ali3 1 Faculity of Science, University of Zakho, Duhok , Kurdistan Region-Iraq 2 College of Basic Education, University of Salahaddin, Erbil, Kurdistan Region-Iraq 3 College of Computer Science and Mathematics, University of Al Mosul, Mosul, Iraq Abstract. One of the more exciting polynomials among the newly presented graph algebraic poly- nomials is theM−Polynomial, which is a standard method for calculating degree−based topological indices. In this paper, we define the Mve−polynomials based on vertex–edge degree and derive various vertex–edge degree based topological indices from them. Thus, for any graph, we provide some relationships between vertex–edge degree topological indices. Also, we discuss the general Mve−polynomial of r−regular simple graph. Finally, we computed the Mve−polynomial of the 2−ary tree graph. 2020 Mathematics Subject Classifications: 05C05, 05C07, 05C10, 94C15 Key Words and Phrases: Mve−polynomials, Mve−indices, regular graph, 2−ary tree 1. Introduction Let G be a connected simple graph and let w be a weight given to its vertices. That is, for each v ∈ V (G) ”V = V (G) be the vertex set”, w(v) is a positive integer. A Wve−polynomial of G is defined by: Wve(G;x, y) = Σuv∈E(G)x w(u)yw(v), w(u) ≤ w(v), (1.1) where ”E = E(G) be the edge set”. Collecting all similar terms xiyj we can rewrite this polynomial as: Wve(G;x, y) = Σuv∈E(G)nijx iyj , i ≤ j, (1.2) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4698 Email addresses: kavi.rasool@uoz.edu.krd (K. Rasool), payman.rashed@su.edu.krd (P. Rashed), ahmedgraph@uomosul.edu.iq (A. Ali) https://www.ejpam.com 773 © 2023 EJPAM All rights reserved. K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 774 where nij is the number of all edges uv such that w(u) = i and w(v) = j. The degree of a vertex u ∈ V is the number of edges incident on u, denoted as du. The neighborhood of a vertex u ∈ V,NG(u) is a set of all neighbors of u, i.e., NG(u) = {v|uv ∈ E(G)} and its called open neighborhood. The closed neighborhood of a vertex u, denoted by NG[u], is obtained by adding a vertex u to NG(u), that is, NG[u] = NG(u) ∪ {u}. The δu denotes the degrees sum of neighbors of u in G. It is clear that (1.1) is a general Wve−polynomial. If w(u) = du for all u ∈ V (G), then (1.1) is M−polynomial of G, which may simplified as: M(G;x, y) = Σuv∈E(G)mijx iyj , i ≤ j, (1.3) where mij is the number of edges uv ∈ E(G) such that {du, dv} = {i, j}. This polynomial was first introduced by Deutsch and Klavžar [6]. And, if w(u) = δu for all u ∈ V (G), then from (1.1), we get NM−polynomial of G: NM(G;x, y) = Σuv∈E(G)m ′ ijx iyj , i ≤ j, (1.4) where m′ ij is the total number of edges uv ∈ E(G) such that {δu, δv} = {i, j}. Mondal and others [13] developed a M−polynomial into a NM−polynomial of a graph G. Now, for each u ∈ V (G), τu is defined as the number of all edges in G incident to a vertex of NG[u]. From (1.1), substituting w(u) = τu, we get Mve−polynomial: Mve(G) = Σuv∈E(G)x τuyτv , τu ≤ τv, (1.5) Simplifying (1.5) by collecting similar terms, we get Mve(G) = Σuv∈E(G)cijx τuyτv , τu ≤ τv, in which cij is the number of all edges uv ∈ E(G) such that {τu, τv} = {i, j}. The terms vertex–edge degree of the graphs, τu were first introduced by Chellali and others [4]. The authors defined these novel degree concepts in relation to the vertex−edge domination parameters [4, 10]. The vertex–edge degree (ve−degree) concepts of the graphs were extensively used in chemical graph theory [3, 7, 8] and [19]. The M−polynomial is the most general polynomial that may generate a wide range of degree−based topological indices [5, 6, 9, 11–13, 15, 17] and [18]. K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 775 From the above three definitions, we get some properties: (i) Σi≤jmijx iyj |x=y=1 = Σi≤jnijx iyj |x=y=1 = Σi≤jcijx iyj |x=y=1 = q. (ii) For any vertex u in a graph G, we have: du ≤ τu ≤ δu. (iii) Let f(hu, hv) be the index function, where hz ∈ {dz, τz, δz}. Then f(du, dv) ≤ f(τu, τv) ≤ f(δu, δv), where f ∈ [first and second Zagreb, reduced first and second Zagreb, hyper Zagreb index, forgotten index, albertson index, sigma index]. Finally, there are many polynomials that have been found over the current century that have chemical applications; see [1, 2] and [14] 2. Some Relations Between Vertex – Edge Degree Topological Indices In this section, we give some relations between vertex–edge degree topological indices for any graph G. Also, the lower and upper bounds for vertex–edge degree topological indices via Wve−polynomial are determined; some of them are shown in Table 1. for any graph G. Table 1: Some vertex–edge–degree based topological indices for Wve−polynomial. Topological index Symbol Index Formula f(w(u), w(v)) Derivation from Wve(G;x, y) First Zagreb W 1 ve(G) Σuv∈E(G)(w(u) + w(v)) (Dx +Dy)Wve(G;x, y)|x=y=1 Second Zagreb W 2 ve(G) Σuv∈E(G)(w(u)w(v)) (DxDy)Wve(G;x, y)|x=y=1 Reduced First Zagreb RW 1 ve(G) Σuv∈E(G)(w(u) + w(v)− 2) (Dx +Dy − 2)Wve(G;x, y)|x=y=1 Reduced Second Zagreb RW 2 ve(G) Σuv∈E(G)(w(u)− 1)(w(v))− 1) (Dx − 1)(Dy − 1)Wve(G;x, y)|x=y=1 Hyper Zagreb index HypWve(G) Σuv∈E(G)(w(u) + w(v))2 D2 xJWve(G;x, y)|x=y=1 Forgotten index FWve(G) Σuv∈E(G)((w(u)) 2 + (w(v))2) (D2 x +D2 y)Wve(G;x, y)|x=y=1 Albertson index AlbWve(G) Σuv∈E(G)|w(u)− w(v)| DxIWve(G;x, y)|x=y=1 Sigma index σWve(G) Σuv∈E(G)(w(u)− w(v))2 D2 xIWve(G;x, y)|x=y=1 Where the operator Dx and Dy on Wve(G;x, y) are defined as: DxWve(G;x, y) = x ∂Wve(G;x, y) ∂x ,DyWve(G;x, y) = y ∂Wve(G;x, y) ∂y , JWve(G;x, y) = Wve(G;x, x) and IWve(G;x, y) = Wve(G;x, x−1). Theorem 2.1: Let G be any graph with order p = |V (G)| and size q = |E(G)|. Then 1. RW 1 ve(G) = W 1 ve(G)− 2q. 2. RW 2 ve(G) = W 2 ve(G)−W 1 ve(G) + q. 3. FWve(G) = HypWve(G)− 2W 2 ve(G). 4. σWve(G) = HypWve(G)− 4W 2 ve(G). 5. σWve(G) = 2FWve(G)−HypWve(G). K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 776 Proof: 1. From the definition of the reduced first Zagreb, we have: RW 1 ve(G) = Σuv∈E(G)(w(u) + w(v)− 2) = Σuv∈E(G)(w(u) + w(v))− Σuv∈E(G)2 = W 1 ve(G)− 2q. 2. From the definition of the reduced second Zagreb, we have: RW 2 ve(G) = Σuv∈E(G)(w(u)− 1)(w(v)− 1) = Σuv∈E(G)w(u)w(v)− Σuv∈E(G)(w(u) + w(v)) + Σuv∈E(G)1 = R2 ve(G)−R1 ve(G) + q. 3. From the definition of the Forgotten index, we have: FWve(G) = Σuv∈E(G)(w(u) 2 + w(v)2) = Σuv∈E(G)(w(u) + w(v))2 − 2Σuv∈E(G)w(u)w(v) = HypWve(G)− 2W 2 ve(G). 4. From the definition of the sigma index, we have: σWve(G) = Σuv∈E(G)(w(u)− w(v))2 = Σuv∈E(G)((w(u)) 2 + (w(v))2)− 2Σuv∈E(G)w(u)w(v) = FWve(G)− 2W 2 ve(G) = HypWve(G)− 4W 2 ve(G). 5. From 3 and 4, we have: σWve(G) = 2FWve(G)−HypWve(G). . In the next theorem, the upper and lower boundaries will be found using the vertex−degree−based topological indices τu, u ∈ V (G). Theorem 2.2: Let G be any graph with order p and size q. Then 1. 2q ≤ M1 ve(G) ≤ 2q2. 2. q ≤ M2 ve(G) ≤ q3. 3. 0 ≤ RM1 ve(G) ≤ 2q(q − 1). 4. 0 ≤ RM2 ve(G) ≤ q(q − 1)2. 5. 4q ≤ HypMve(G) ≤ 4q3. K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 777 6. 2q ≤ FMve(G) ≤ 2q3. Proof: Since 1 ≤ τu ≤ q, for all u in G, then 1. 2 ≤ τu + τv ≤ 2q , this implies that Σuv∈E(G)2 ≤ Σuv∈E(G)(τu + τv) ≤ Σuv∈E(G)2q, then 2q ≤ M1 ve(G) ≤ 2q2. 2. 1 ≤ τuτv ≤ q2, this implies that Σuv∈E(G)1 ≤ Σuv∈E(G)(τuτv) ≤ Σuv∈E(G)q 2, then q ≤ M2 ve(G) ≤ q3. 3. 0 ≤ τu + τv − 2 ≤ 2(q − 1) , this implies that Σuv∈E(G)0 ≤ Σuv∈E(G)(τu + τv − 2) ≤ Σuv∈E(G)2(q − 1), then 0 ≤ RM1 ve(G) ≤ 2q(q − 1). 4. 0 ≤ (τu − 1)(τv − 1) ≤ (q − 1)2, this implies that Σuv∈E(G)0 ≤ Σuv∈E(G)(τu − 1)(τv − 1) ≤ Σuv∈E(G)(q − 1)2, then 0 ≤ RM2 ve(G) ≤ q(q − 1)2. 5. 4 ≤ (τu + τv) 2 ≤ 4q2, this implies that Σuv∈E(G)4 ≤ Σuv∈E(G)(τu + τv) 2 ≤ Σuv∈E(G)4q 2, then 4q ≤ HypMve(G) ≤ 4q3. 6. 2 ≤ (τu) 2 + (τv) 2 ≤ 2q2, this implies that Σuv∈E(G)2 ≤ Σuv∈E(G)((τu) 2 + (τv) 2) ≤ Σuv∈E(G)2q 2, then 2q ≤ FMve(G) ≤ 2q3. Remark: We note that equality exists for all topological indices in Theorem 2.2, if G = K2. 3. Mve−Polynomials of r−Regular Graphs In the next theorems, we discuss the general Mve−polynomial of r−regular simple graph of size q. Theorem 3.1: Let G is an r−regular simple graph G of size q, then K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 778 1. M(G;x, y) = qxryr. 2. NM(G;x, y) = qxr 2 yr 2 . 3. If the graph G without triangle cycles, then Mve(G;x, y) = qxr 2 yr 2 . Proof: 1. Let G is r−regular graph with q edges, then every edge in G is an incident on two vertices which have r degree. Hence, M(G;x, y) = qxryr. 2. Since every vertex u in r−regular graph G is adjacent r vertices, then δu = r2. Hence, NM(G;x, y) = qxr 2 yr 2 . 3. If G is an r−regular simple graph without triangle cycles, then τu = τv = r2, for every edge e = uv where u, v ∈ V (G). Hence Mve(G;x, y) = qxr 2 yr 2 . Example 3.2: Let Hi be 3−regular graph for all i = 1, 2, 3. See Figure 1. Figure 1: 3−regular graphs. We note that: M(H1;x, y) = 9x3y3, NM(H1;x, y) = 9x9y9 and Mve(H1;x, y) = 9x8y8. M(H2;x, y) = 12x3y3 and NM(H2;x, y) = Mve(H2;x, y) = 12x9y9. M(H3;x, y) = 15x3y3, NM(H3;x, y) = 15x9y9 and Mve(H3;x, y) = 7x8y8 + 4x8y9 + 4x9y9. It is very difficult to find a general formula for Mve−polynomial of a regular graph, so some conditions were given on the regular graphs in order to obtain Mve−polynomial. Theorem 3.3: Let G is an r−regular simple graph such that every vertex in G lies on only one triangle cycle, then K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 779 Mve(G;x, y) = qxr 2−1yr 2−1, where q = |E(G)|. Proof: Let e = uv be any edge in G where u, v ∈ V (G), then the two vertices u and v are adjacent with third vertex, then τu = τv = r2 − 1. Hence Mve(G;x, y) = qxr 2−1yr 2−1, where q = |E(G)|. Theorem 3.4: Let G is a r−regular simple graph such that any vertex in G lies at most on one triangular cycle. If h be the number of edges lies between any two vertices belong to triangular cycle (or cycles) and k be the number of edges which one of whose ends, but not both lies on a triangular cycle. Then Mve(G;x, y) = hxr 2−1yr 2−1 + kxr 2−1yr 2 + (q − h− k)xr 2 yr 2 , where q = |E(G)|. Proof: Obvious. Definition 3.5: Let G be a simple graph and a vertex u ∈ V (G), we can rewrite δu as: δu = Σz∈N(u)dz = du + 2ε1 + ε2, where ε1 be the number of edges of which both its ends belong to a triangular cycle and ε2 be the number of edges which lies on the vertices are neighbors of a vertex u. Theorem 3.6: Let G be a any graph without triangular cycle, then, NM(G;x, y) = Mve(G;x, y). Proof: For edge e = uv where u, v ∈ V (G), then m′ δuδv xδuyδv = m′ δuδv xdu+2ε1+ε2ydv+2ε1+ε2 , by Definition 3.5. Since G is a graph without triangular cycles, then ε1 = 0. Hence m′ δuδv xδuyδv = m′ δuδv xdu+ε2ydv+ε2 = cδuδvx τuyτv , by definition vertex – edge degree of the graph. Hence, NM(G;x, y) = Mve(G;x, y). 4. Mve−Polynomials of 2−Ary Tree Graph Definition 4.1: [16] A rooted tree G is an acyclic connected graph with a special node that is called the root of the tree and every edge directly or indirectly originates from the K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 780 root. An ordered rooted tree is a rooted tree where the children of each internal vertex is ordered. If every internal vertex of a rooted tree has not more than m children, it is called an m−ary tree. In this section, we determine the Mve−polynomial of special case of m−ary tree is 2−ary tree of n levels denoted by ℵn and as shown in Figure 2. Figure 2: 2−Ary Tree graph ℵn. Some Properties of a 2−Ary Tree Graph ℵn: • At each level of i, the number of vertices are 2i, for 0 ≤ i ≤ n. • The order and the size are p(ℵn) = 2n+1 − 1 and q(ℵn) = 2n+1 − 2, respectively. • The rooted vertex of degree 2, the degrees of vertices at each level of i, 1 ≤ i ≤ n−1 are 3 which represent the maximum degree ” △ (ℵn) = 3” and the degrees of the last level are 1 which represent the minimum degree ”δ(ℵn) = 1”. • The maximum and minimum ve−degree are △ve (ℵn) = 9 and δve(ℵn) = 3, respec- tively. Theorem 4.2: Let ℵn be the 2−ary tree graph of order 2n+1 − 1 , n ≥ 4. Then, Mve(ℵn;x, y) = 2x6y8 + 4x8y9 + 2nx3y5 + 2n−1x5y9 + (2n−1 − 8)x9y9. Proof: From the Definition 4.1 and Figure 2 of 2−ary tree graph ℵn, we can observe that the vertices are divided into five partitions: |V1| = |v ∈ V (ℵn) : τv = 3| = 2n, |V2| = |v ∈ V (ℵn) : τv = 5| = 2n−1, |V3| = |v ∈ V (ℵn) : τv = 6| = 20, |V4| = |v ∈ V (ℵn) : τv = 8| = 21, |V5| = |v ∈ V (ℵn) : τv = 9| = 2n−1 − 4. K. Rasool, P. Rashed, A. Ali / Eur. J. Pure Appl. Math, 16 (2) (2023), 773-783 781 The edges set of 2 − ary tree graph ℵn can be partitioned as |E3,5| = |uv ∈ E(ℵn) : τu = 3 and τv = 5| = 2n, |E5,9| = |uv ∈ E(ℵn) : τu = 5 and τv = 9| = 2n−1, |E6,8| = |uv ∈ E(ℵn) : τu = 6 and τv = 8| = 2, |E8,9| = |uv ∈ E(ℵn) : τu = 8 and τv = 9| = 22, |E9,9| = |uv ∈ E(ℵn) : τu = τv = 9| = |E(ℵn)| − |E3,5| − |E5,9| − |E6,8| − |E8,9| = 2n−1 − 8. Thus, the Mve−polynomial of 2−ary tree graph ℵn is Mve(ℵn;x, y) = Σi≤jcijx iyj = Σ3≤5c35x 3y5 +Σ5≤9c59x 5y9 +Σ6≤8c68x 6y8 +Σ8≤9c89x 8y9 +Σ9≤9c99x 9y9 = Σuv∈E3,5c35x 3y5+Σuv∈E5,9c59x 5y9+Σuv∈E6,8c68x 6y8+Σuv∈E8,9c89x 8y9+Σuv∈E9,9c99x 9y9. = |E3,5|x3y5 + |E5,9|x5y9 + |E6,8|x6y8 + |E8,9|x8y9 + |E9,9|x9y9 = 2nx3y5 + 2n−1x5y9 + 2x6y8 + 4x8y9 + (2n−1 − 8)x9y9. Corollary 4.3: Let ℵn be the 2− ary tree graph of order 2n+1 − 1 , n ≥ 4. Then 1. M1 ve(ℵn) = 8(2n+1 + 2n − 6). 2. M2 ve(ℵn) = 3(42× 2n−1 + 5× 2n − 88). 3. RM1 ve(ℵn) = (2n+4 + 2n+3 − 2n+2 − 44). 4. RM2 ve(ℵn) = 2(2n+2 + 48× 2n−1 − 109). 5. HypMve(ℵn) = 4(2n+4 + 130× 2n−1 − 261). 6. FMve(ℵn) = 2(17× 2n + 134× 2n−1 − 258). 7. AlbMve(ℵn) = 4(2n + 2). 8. σMve(ℵn) = 4(2n+1 + 2n + 3). Remark 4.4: From Theorem 3.5, we get Mve(ℵn;x, y) = NM(ℵn;x, y) = 2x6y8 + 4x8y9 + 2nx3y5 + 2n−1x5y9 + (2n−1 − 8)x9y9. 5. Conclusion: In this paper, we have given a new polynomial based on the terms of a vertex–edge degree of a graph G. From this polynomial we proved many properties and proved that it’s equal to the neighbor polynomial when G there is a graph without a triangular cycle. We also discover a vertex−edge degree polynomial for an r−regular graph under certain conditions. REFERENCES 782 References [1] Haveen J Ahmed, Ahmed M Ali, and Gashaw A Mohammed Saleh. Detour polynomi- als of vertex coalenscence and bridges coalenscence graphs. Asian-European Journal of Mathematics, 15(02):2250025, 2022. [2] Ahmed Mohammed Ali, Herish Omer Abdullah, and Gashaw Aziz Mohammed Saleh. Hosoya polynomials and wiener indices of carbon nanotubes using mathematica pro- gramming. Journal of Discrete Mathematical Sciences and Cryptography, 25(1):147– 158, 2022. [3] Murat Cancan, Süleyman Ediz, Mehdi Alaeiyan, and Mohammad Reza Farahani. On ve-degree molecular properties of copper oxide. Journal of Information and Opti- mization Sciences, 41(4):949–957, 2020. [4] Mustapha Chellali, Teresa W Haynes, Stephen T Hedetniemi, and Thomas M Lewis. On ve-degrees and ev-degrees in graphs. Discrete Mathematics, 340(2):31–38, 2017. [5] Yu-Ming Chu, Mehwish Hussain Muhammad, Abdul Rauf, Muhammad Ishtiaq, and Muhammad Kamran Siddiqui. Topological study of polycyclic graphite carbon ni- tride. Polycyclic Aromatic Compounds, 42(6):3203–3215, 2020. [6] Emeric Deutsch and Sandi Klavžar. M-polynomial and degree-based topological in- dices. arXiv preprint arXiv:1407.1592, 2014. [7] Süleyman Ediz. Predicting some physicochemical properties of octane isomers: A topological approach using ev-degree and ve-degree zagreb indices. arXiv preprint arXiv:1701.02859, 2017. [8] VR Kulli. On ve-degree indices and their polynomials of dominating oxide networks. Annals of Pure and Applied Mathematics, 18(1):1–7, 2018. [9] Young Chel Kwun, Mobeen Munir, Waqas Nazeer, Shazia Rafique, and Shin Min Kang. M-polynomials and topological indices of v-phenylenic nanotubes and nanotori. Scientific reports, 7(1):1–9, 2017. [10] Jason Robert Lewis. Vertex-edge and edge-vertex parameters in graphs. PhD thesis, Clemson University, 2007. [11] Sourav Mondal, Muhammad Imran, Nilanjan De, and Anita Pal. Neighborhood m- polynomial of titanium compounds. Arabian Journal of Chemistry, 14(8):103244, 2021. [12] Sourav Mondal, DE Nilanjan, and PAL Anita. Topological properties of networks using m-polynomial approach. Konuralp Journal of Mathematics, 8(1):97–105, 2020. [13] Sourav Mondal, Muhammad Kamran Siddiqui, Nilanjan De, and Anita Pal. Neigh- borhood m-polynomial of crystallographic structures. Biointerface Res. Appl. Chem, 11(2):9372–9381, 2021. REFERENCES 783 [14] Raghad A Mustafa, Ahmed M Ali, and AbdulSattar M Khidhir. Mn–polynomials of some special for cog-graphs. Journal of Discrete Mathematical Sciences and Cryptog- raphy, pages 1–16, 2022. [15] Zahid Raza and Mark Essa K. Sukaiti. M-polynomial and degree based topological indices of some nanostructures. Symmetry, 12(5):831, 2020. [16] Charles Semple and Mike Steel. A supertree method for rooted trees. Discrete Applied Mathematics, 105(1-3):147–158, 2000. [17] Ashish Verma, Sourav Mondal, Nilanjan De, and Anita Pal. Topological properties of bismuth tri-iodide using neighborhood m-polynomial. International Journal of Mathematics Trends and Technology, 67(10):83–90, 2019. [18] Satyanarayana Vollala and Indrajeet Saravanan. Vertex degree-based topological in- dices of penta-chains using m-polynomial. International Journal of Advances in En- gineering Sciences and Applied Mathematics, 11(1):53–67, 2019. [19] Jing Zhang, Muhammad Kamran Siddiqui, Abdul Rauf, and Muhammad Ishtiaq. On ve-degree and ev-degree based topological properties of single walled titanium dioxide nanotube. Journal of Cluster Science, 32(4):821–832, 2021.