International Journal of Analysis and Applications Volume 19, Number 2 (2021), 205-227 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-205 DOWNHILL ZAGREB TOPOLOGICAL INDICES OF GRAPHS BASHAIR AL-AHMADI, ANWAR SALEH∗, WAFA AL-SHAMMAKH Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia ∗Corresponding author: asaleh1@uj.edu.sa Abstract. Topological indices are graph invariants determined by the distance or degree of vertices of the molecular graph. Topological indices have been used effectively in chemical graph theory in explaining the structures and predicting certain physicochemical properties of chemical compounds. In this research, we introduce the first, second, and forgotten downhill Zagreb indices and calculate those topological indices for some standard families of graphs and the join of graphs. Also, the downhill topological indices for the firefly graph, book graph, and stacked book graph are established. Finally, the downhill indices of Graphene and honeycomb network are obtained. 1. Introduction A graph is non empty set of vertices together with a number of edges connecting a subset of them. V (G) and E(G) denoted for vertex set and edge set respectively. If we consider the molecules as special chemical structures, and if we replace atoms and bonds with vertices and edges, respectively, the obtained graph is called a molecular graph. That means a molecular graph is a simple graph such that its vertices correspond to the atoms and its edges to the bonds. We note that hydrogen atoms are often omitted and the remaining part of the graph is sometimes called as the carbon graph of the corresponding molecule. Chemical graph theory which deals with the above mentioned relations between molecules and correspond- ing graphs is a branch of mathematical chemistry which has an important effect on the development of Received December 24th, 2020; accepted January 25th, 2021; published February 12th, 2021. 2010 Mathematics Subject Classification. 05C35, 05C07, 05C40. Key words and phrases. first downhill Zagreb index (of a graph); second downhill Zagreb index; forgetten downhill Zagreb index. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 205 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-205 Int. J. Anal. Appl. 19 (2) (2021) 206 the molecular chemistry along with quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies. In this research work, we are concerned with simple graphs which they are finite, undirected with no loops and multiple edges. The open and closed neighborhoods of a vertex v in a graph G are denoted by N(v) = {u ∈ V : uv ∈ E} and N[v] = N(v) ∪{v}, respectively. The degree of a vertex v in G is denoted by degG(v) or briefly by dG(v), where degG(v) = |N(v)|. When there is no confusion, one can also omit G and use d(v) instead of dG(v). The minimum degree and maximum degree of G are denoted by δ, and ∆ respectively. In a graph G if δ = ∆ = k, then the graph G is called regular graph of degree k. Also a graph with the property that ∆ ≤ 4 is called a chemical graph. The following notations and different types of graphs well known in the literature [9] and [2]. A Double star is the graph obtained from K2 by joining s pendent edges to one end and r pendent edges to the other end of K2. A wheel Wn+1, n > 3 is the join of Cn and K1. A helm graph, denoted by Hn, is a graph obtained from Wn+1 by attaching an end edge to each rim vertex of Wn+1, where the vertices corresponding to Cn are known as rim vertices. The gear graph is a wheel graph with a vertex added between each pair adjacent graph vertices of the outer cycle. The gear graph Gn has 2n + 1 vertices and 3n edges. The sierpinski sieve graph Sn is the graph obtained from the connectivity of the sierpinski sieve. The graph has 3(3n−1+1) 2 vertices and 3n edges. The tadpole graph Tr,s is obtained by joining a cycle Cr and a path Ps by a bridge, where r > 3 and s > 1. The Cartesian product G of two graphs G1 and G2, denoted G1×G2, has vertex set V (G) = V (G1) × V (G2), and two distinct vertices (a,b) and (c,d) of G1 × G2 are adjacent if either a = c and bd ∈ E(G2), or b = d and ac ∈ (G1). The join G = G1 ∨G2 of two graphs G1 and G2 has vertex set V (G) = V (G1) ⋃ V (G2) and edge set E(G) = E(G1) ⋃ E(G2){uv : u ∈ V (G1),v ∈ V (G2)}. The corona product G◦H is defined as the graph obtained from G and H by taking one copy of G and n1 copies of H and joining by an edge each vertex from the ith-copy of H with the ith-vertex of G. Book graph is a Cartesian product of a star and single edge, denoted by Bm. The m-book graph is defined as the graph Cartesian product Sm+1 ×P2 , where Sm+1 is a star graph and P2 is the path graph . The stacked book graph of order (m,n) is defined as the graph Cartesian product Sm+1 ×Pn, where Sm is a star graph and Pn is the path graph on n nodes, and it is denoted by Bm,n. As it is known, the path, cycle, and complete graphs with n vertices are denoted by Pn, Cn, Kn; and the complete bipartite graph is denoted by Kr,s. Int. J. Anal. Appl. 19 (2) (2021) 207 A topological index of a graph is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In recent decades, a large number of topological indices have been defined and utilized for chemical documenta- tion, isomer discrimination, study of molecular complexity, chirality, similarity/dissimilarity, QSAR/QSPR, drug design and database selection, lead optimization, etc. As an example, the boiling point of a molecule is directly related to the forces between the atoms. When a solution is heated, the temperature is increased and as it is increased, the kinetic energy between molecules increases. This means that the molecular motion becomes so intense that the bonds between molecules break and become a gas. The moment the liquid turns to gas is labeled as the boiling point. The boiling point can give important clues about the physical properties of chemical structures. Molecules which strongly interact or bond with each other through a variety of inter-molecular forces cannot move easily or rapidly and therefore, do not achieve the kinetic energy necessary to escape the liquid state. That is why the boiling points of the alkanes increase with molecular size. The most useful and famous topological indices of a graph are the first and second Zagreb indices which have been introduced by Gutman and Trinajstic in [7]. They are denoted by M1(G) and M2(G) and were defined as M1(G) = ∑ u∈V (G) [ d(u) ]2 and M2(G) = ∑ uv∈E(G) d(u)d(v), respectively. The forgotten topological index was introduced by Furtula and Gutman [4] and it is stated that the forgotten topological index is a special case of first Zagreb index. Shehnaz Akhter, Muhammad Imran extended the study of forgotten topological index and determined the close formula of F-index for some graph operations. The Zagreb indices have been studied extensively. Many new reformulated and extended versions of the Zagreb indices have been introduced. For more discussion on these indices, we encourage the readers to consult the articles ( [1], [3], [5], [6], [11], [12], [13], [14], [16], [17], [18], [19]). The downhill domination is introduced for the first time in [10]. For more details about the downhill and uphill domination, we refer to [8, 15]. In this research work, motivated by the downhill domination and the huge applications of topological indices, we define the downhill degree and introduce the first, second and forgotten downhill Zagreb indices Int. J. Anal. Appl. 19 (2) (2021) 208 and calculate these topological indices for, some standard families of graphs, join of two graph. Also, the downhill topological indices for firefly graph, book graph Stacked book graph are established. Finally, the downhill indices of Graphene and Honeycomb Network are obtained. 2. Downhill Zagreb Indices In this section, we define the first, second and forgotten downhill Zagreb indices and calculate these topological indices for some standard families of graphs. Definition 2.1. [10] Let G = (V,E) be a graph. A u−v path P in G is a sequence of vertices in G, starting with u and ending at v, such that consecutive vertices in P are adjacent, and no vertex is repeated. A path π = v1,v2, ...vk+1 in G is a downhill path if for every i,1 6 i 6 k, deg(vi) > deg(vi+1). Definition 2.2. In a graph G = (V,E). A vertex v is downhill dominates a vertex u if there exists a downhill path originated from v to u. The downhill neighborhood of a vertex v is denoted by Ndn(v) and defined as: Ndn(v) = {u : v downhill dominates u} . The downhill degree of the vertex v, denotes by ddn(v), is the number of downhill neighbors of v, that means ddn(v) = |Ndn(v)|. The downhill closed neighborhood, Ndn[v], of a vertex v is the downhill open neighborhood of v taken together with v. It follows that Ndn[v] = Ndn(v) ∪{v}. Definition 2.3. Let G = (V,E) be a graph. Then the first, second and forgotten downhill Zagreb indices are defined as: DWM1(G) = ∑ v∈V (G) (ddn(v)) 2, DWM2(G) = ∑ vu∈E(G) ddn(v)ddn(u) and DWF(G) = ∑ v∈V (G) (ddn(v)) 3. Example 2.1. Let G be a house graph as in Figure 1. Then DWM1(G) = ∑ v∈V (G) (ddn(v)) 2 = 0 + 25 + 25 + 25 + 1 + 1 = 77, DWM2(G) = ∑ vu∈E(G) ddn(v)ddn(u) = 0 × 5 + 3 × 5 × 5 + 2 × 5 × 1 + 1 × 1 = 86, DWF(G) = ∑ v∈V (G) (ddn(v)) 3 = 0 + 125 + 125 + 125 + 1 + 1 = 377. Int. J. Anal. Appl. 19 (2) (2021) 209 Figure 1. The House Graph. Proposition 2.1. Let G be the connected k-regular graph of n vertices. Then, DWM1(G) = n(n− 1)2, DWM2(G) = nk(n− 1)2 2 , DWF(G) = n(n− 1)3. Proof. Let G be the connected k-regular graph of n vertices. It is easy to see that for any vertex in G has downhill degree n− 1. Hence, DWM1(G) = n(n− 1)2 and DWF(G) = n(n− 1)3. Similarly, there are nk2 edges in G, where each edge has endpoints of downhill degree n− 1. Hence, DWM2(G) = nk(n−1)2 2 . � Corollary 2.1. (1) For any cycle Cn, DWM1(Cn) = DWM2(Cn) = n(n− 1)2 and DWF(Cn) = n(n− 1)3. (2) For any complete graph Kn, DWM1(Kn) = n(n − 1)2, DWM2(Kn) = n(n−1)3 2 and DWF(Kn) = n(n− 1)3. Proposition 2.2. Let G ∼= Pn be a path of n ≥ 3 vertices. Then DWM1(G) = (n− 2)(n− 1)2, DWM2(G) = (n− 3)(n− 1)2, DWF(G) = (n− 2)(n− 1)3. Proof. Let G ∼= Pn be a path of n vertices, where n ≥ 3, the path has n−2 vertices of downhill degree n−1 and 2 vertices of downhill degree 0. Hence, DWM1(G) = (n−2)(n−1)2 and DWF(G) = (n−2)(n−1)3. There are n−1 edges in G of which n−3 edges have endpoints of downhill degree n−1 and 2 edges have one endpoint of downhill degree n−1 and the another endpoint of downhill degree 0. Hence, DWM2(G) = (n−3)(n−1)2. � Int. J. Anal. Appl. 19 (2) (2021) 210 Proposition 2.3. Let G ∼= Sn be a star with n + 1 vertices, where n ≥ 2. Then, DWM1(G) = n 2, DWM2(G) = 0, DWF(G) = n3. Proof. Let G ∼= Sn be a star with n + 1 vertices, where n ≥ 2, clearly there are n + 1 vertices of downhill degree n and n vertices of downhill degree 0. Hence,DWM1(G) = n 2 and DWF(G) = n3. The star graph G has n edges, where each edge has one endpoint of downhill degree n and the another endpoint of downhill degree 0. Hence, DWM2(G) = 0. � Proposition 2.4. Let G ∼= Ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. Then, DWM1(G) =   2(s + r + 1)2 if s = r; (s + r + 1)2 + r2 if s > r. Proof. Let G ∼= Ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. We have two cases: Case 1. If s = r. Then the graph has two vertices of downhill degree s + r + 1 and s + r vertices of downhill degree 0. Hence, DWM1(G) = 2(s + r + 1) 2. Case 2. If s > r, then there are one vertex of downhill degree s + r + 1, one vertex of downhill degree r and s + r vertices of downhill degree 0. Hence, DWM1(G) = (s + r + 1) 2 + r2. � Proposition 2.5. Let G ∼= Ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. Then, DWM2(G) =   (s + r + 1)2 if s = r; r(s + r + 1) if s > r. Proof. Let G ∼= Ss,r be a double star with s + r + 2 vertices and s + r + 1 edges, where s,r ≥ 2. We have two cases: Case 1. If s = r. It has one edge has endpoints of downhill degree s + r + 1. Also, there are s + r edges in G, where each edge has one endpoint of downhill degree s + r + 1 and the another endpoint of downhill degree 0. Hence, DWM2(G) = (s + r + 1) 2. Case 2. If s > r. It has one edge has one endpoint of downhill degree s + r + 1 and the another endpoint of downhill degree r. Also, there are s + r edges where each edge has endpoint of downhill degree s + r + 1 or r and the another endpoint of downhill degree 0. Hence, DWM2(G) = r(s + r + 1). � In the same way we can get the following result. Int. J. Anal. Appl. 19 (2) (2021) 211 Proposition 2.6. Let G ∼= Ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. Then, DWF(G) =   2(s + r + 1)3 if s = r; (s + r + 1)3 + r3 if s > r. Proposition 2.7. Let G ∼= Ks,r be the complete bipartite graph, where s < r. Then, DWM1(G) = sr 2, DWM2(G) = 0, DWF(G) = sr3. Proof. Let G ∼= Ks,r be the complete bipartite graph, where s < r. There are s+r vertices of which s vertices of downhill degree r and r vertices of downhill degree 0. Hence, DWM1(G) = sr 2 and DWF(G) = sr3. There are sr edges in G, where each edge has one endpoint of downhill degree r and the another endpoint of downhill degree 0. Hence, DWM2(G) = 0. � Proposition 2.8. Let G ∼= Wn be a wheel graph of n + 1 vertices, where n ≥ 4. Then, DWM1(G) = n(n 2 −n + 1), DWM2(G) = n(2n 2 − 3n + 1), DWF(G) = n(n3 − 2n2 + 3n− 1). Proof. Let G ∼= Wn be a wheel graph of n + 1 vertices, where n ≥ 4. There is one vertex of downhill degree n and n vertices of downhill degree n − 1. Hence, DWM1(G) = n(n2 − n + 1) and DWF(G) = n(n3 −2n2 + 3n−1). There are 2n edges in G of which n edges have one endpoint of downhill degree n and the another endpoint of downhill degree n− 1 and n edges have endpoints of downhill degree n− 1. Hence, DWM2(G) = n(2n 2 − 3n + 1). � Proposition 2.9. Let G ∼= Gn be a gear graph of 2n + 1 vertices, where n ≥ 4. Then, DWM1(G) = 4n(n + 1), DWM2(G) = 4n 2, DWF(G) = 8n(n2 + 1). Proof. Let G ∼= Gn be a gear of 2n + 1 vertices, where n ≥ 4. Then the graph has one vertex of downhill degree 2n, n vertices of downhill degree 2 and n vertices of downhill degree 0. Hence, DWM1(G) = 4n(n+ 1) and DWF(G) = 8n(n2 + 1). There are 3n edges in G of which n edges have one endpoint of downhill degree 2n and the another endpoint of downhill degree 2 and 2n edges have one endpoint of downhill degree 2 and the another endpoint of downhill degree 0. Hence, DWM2(G) = 4n 2. � Int. J. Anal. Appl. 19 (2) (2021) 212 Proposition 2.10. Let G ∼= Hn be a helm graph of 2n + 1 vertices, where n ≥ 5. Then, DWM1(G) = n(4n 2 + 1), DWM2(G) = n(8n 2 − 6n + 1), DWF(G) = n(8n3 − 4n2 + 6n− 1). Proof. Let G ∼= Hn be a helm graph of 2n + 1 vertices, where n ≥ 5. Then there are one vertex of downhill degree 2n, n vertices of downhill degree 2n − 1 and n vertices of downhill degree 0. Hence, DWM1(G) = n(4n 2 + 1) and DWF(G) = n(8n3 − 4n2 + 6n − 1). There are 3n edges of which n edges have one endpoint of downhill degree 2n and the another endpoint of downhill degree 2n− 1, n edges have endpoints of downhill degree 2n−1 and n edges have one endpoint of downhill degree 2n−1 and the another one of downhill degree 0. Hence, DWM2(G) = n(8n 2 − 6n + 1). � Proposition 2.11. Let G ∼= Sn be the sierpinski of m vertices, where m = 3(3n−1+1) 2 . Then, DWM1(G) = (m− 3)(m− 1)2, DWM2(G) = (2m− 9)(m− 1)2, DWF(G) = (m− 3)(m− 1)3. Proof. Let G ∼= Sn be the sierpinski graph of m vertices, where m = 3(3n−1+1) 2 . There are m − 3 vertices of downhill degree m − 1 and 3 vertices of downhill degree 0. Hence, DWM1 = (m − 3)(m − 1)2 and DWF(G) = (m − 3)(m − 1)3. There are 2m − 3 edges of which 2m − 9 has endpoints of downhill degree m− 1 and 6 edges have one endpoint of downhill degree m− 1 and the another endpoint of downhill degree 0. Hence, DWM2(G) = (2m− 9)(m− 1)2. � Theorem 2.1. Let G ∼= Tn,m be the tadpole graph with n + m vertices, where n,m ≥ 3. Then, DWM1(G) = (n + m− 1)2 + (n− 1)(n− 2)2 + (m− 1)3, DWM2(G) = (n + m− 1)(2n + m− 5) + (n− 2)3 + (m− 2)(m− 1)2, DWF(G) = (n + m− 1)3 + (n− 1)(n− 2)3 + (m− 1)4. Proof. Let G ∼= Tn,m be the tadpole graph with n + m vertices. Then the graph has one vertex of downhill degree n + m− 1, n− 1 vertices of downhill degree n− 2, m− 1 vertices of downhill degree m− 1 and one vertex of downhill degree 0. Hence, DWM1(G) = (n + m− 1)2 + (n− 1)(n− 2)2 + (m− 1)3, DWF(G) = (n + m− 1)3 + (n− 1)(n− 2)3 + (m− 1)4. Int. J. Anal. Appl. 19 (2) (2021) 213 There are m + n edges of which one edge has one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree m−1, 2 edges have one endpoint of downhill degree n + m−1 and the another endpoint n−2, n−2 edges have endpoints of downhill degree n−2, m−2 edges have endpoints of downhill degree m − 1 and one edge has one endpoint of downhill degree m − 1 and another endpoint of downhill degree 0. Hence, DWM2(G) = (n + m− 1)(2n + m− 5) + (n− 2)3 + (m− 2)(m− 1)2. � 3. The Downhill Zagreb Indices of Graphs under Some Binary Operations Theorem 3.1. Let G ∼= Cn ∨Pm be the join graph with n + m vertices, where n,m ≥ 3. Then, DWM1(G) =   2(n− 1)(2n− 1)2 if n = m; (m− 2)(n + m− 1)2 + (n + 2)(n + 1)2 if n = m + 1; (m− 2)(n + m− 1)2 + 2n2 + n(n− 1)2 if n > m + 1; n(n + m− 1)2 + (m− 2)(m− 1)2 if n < m. Proof. Let G ∼= Cn ∨Pm be the join graph with n + m vertices, where m ≥ 3. We have four cases: Case 1. If n = m. In this case, there are 2n−2 vertices of downhill degree 2n−1 and 2 vertices of downhill degree 0. Hence, DWM1(G) = 2(n− 1)(2n− 1)2. Case 2. If n = m + 1. In this case, there are m−2 vertices of downhill degree n + m−1 and n + 2 vertices of downhill degree n + 1. Hence, DWM1(G) = (m− 2)(n + m− 1)2 + (n + 2)(n + 1)2. Case 3. If n > m + 1. In this case, there are m − 2 vertices of downhill degree n + m − 1, 2 vertices of downhill degree n and n vertices of downhill degree n− 1. Hence, DWM1(G) = (m− 2)(n + m− 1)2 + 2n2 + n(n− 1)2. Case 4. If n < m. In this case, there are n vertices of downhill degree n + m−1, m−2 vertices of downhill degree m− 1 and 2 vertices of downhill degree 0. Hence, DWM1(G) = n(n + m− 1)2 + (m− 2)(m− 1)2. � Int. J. Anal. Appl. 19 (2) (2021) 214 Theorem 3.2. Let G ∼= Cn ∨Pm be the join graph with n + m vertices, where n,m ≥ 3. Then, DWM2(G) =   (n2 − 3)(2n− 1)2 if n = m; A if n = m + 1; B if n > m + 1; C if n < m, where, A = (m− 3)(n + m− 1)2 + 3n(n + 1)2 + (nm− 2n + 2)(n + m− 1)(n + 1), B = (m− 3)(n + m− 1)2 + n(n + m− 1)(nm−m− 2n + 4) + n(n− 1)(3n− 1) and C = n(n + m− 1)2 + (m− 3)(m− 1)2 + n(m− 2)(m− 1)(n + m− 1). Proof. Let G ∼= Cn ∨Pm be the join graph with n + m vertices, where n,m ≥ 3. Clearly the graph G has nm + n + m− 1 edges and we have four cases: Case 1. If n = m. In this case, there are n2−3 edges have endpoints of downhill degree 2n−1 and 2(n + 1) edges have one endpoint of downhill degree 2n− 1 and the another endpoint of downhil degree 0. Hence, DWM2(G) = (n 2 − 3)(2n− 1)2. Case 2. If n = m + 1. In this case, there are m− 3 edges have endpoints of downhill degree n + m− 1, 3n edges have endpoints of downhill degree n + 1 and nm− 2n + 2 edges have one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree n + 1. Hence, DWM2(G) = (m− 3)(n + m− 1)2 + 3n(n + 1)2 + (nm− 2n + 2)(n + m− 1)(n + 1). Case 3. If n > m + 1. In this case, there are m− 3 edges have endpoints of downhill degree n + m− 1, n edges have endpoints of downhill degree n−1, nm−2n edges have one endpoint of downhill degree n+m−1 and the another endpoint of downhill degree n−1, 2n edges have one endpoint of downhill degree n and the another endpoint of downhill degree n− 1 and 2 edges have one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree n. Hence, DWM2(G) = (m− 3)(n + m− 1)2 + n(n + m− 1)(nm−m− 2n + 4) + n(n− 1)(3n− 1). Case 4. If n < m. In this case, there are n edges have endpoints of downhill degree n + m− 1, m− 3 have endpoints downhill degree m− 1, n(m− 2) edges have one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree m−1 and 2(n+ 1) edges have one endpoint of downhill degree n+m−1 or m− 1 and the another endpoint of downhill degree 0. Hence, DWM2(G) = n(n + m− 1)2 + (m− 3)(m− 1)2 + n(m− 2)(m− 1)(n + m− 1). � Int. J. Anal. Appl. 19 (2) (2021) 215 Theorem 3.3. Let G ∼= Cn ∨Pm be the join graph with n + m vertices, where n,m ≥ 3. Then, DWF(G) =   2(n− 1)(2n− 1)3 if n = m; (m− 2)(n + m− 1)3 + (n + 2)(n + 1)3 if n = m + 1; (m− 2)(n + m− 1)3 + 2n3 + n(n− 1)3 if n > m + 1; n(n + m− 1)3 + (m− 2)(m− 1)3 if n < m. Proof. The proof similar to the proof of Theorem 3.1. � Proposition 3.1. Let G ∼= Bm be a book graph of 2(m + 1) vertices, where m ≥ 2. Then, DWM1(G) = 2(4m 2 + 5m + 1), DWM2(G) = m(8m + 7) + 1, DWF(G) = 16m3 + 24m2 + 14m + 2. Proof. Let G ∼= Bm be a book of 2(m + 1) vertices, where m ≥ 2. Then there are 2 vertices of downhill degree 2m + 1 and 2m vertices of downhill degree 1. Hence, DWM1(G) = 2(4m 2 + 5m + 1), DWF(G) = 16m3 + 24m2 + 14m + 2. There are 3m + 1 edges in G of which one edges has endpoints of downhill degree 2m + 1, m edges have endpoints of downhill degree 1 and 2m edges have one endpoint of downhill degree 2m + 1 and the another endpoint of downhill degree 1. Hence, DWM2(G) = m(8m + 7) + 1. � Theorem 3.4. Let G ∼= Bm,t be a stacked book graph with t(m + 1) vertices, where m ≥ 2 and t ≥ 3. Then, DWM1(G) = (t− 2)(t(m + 1) − 1)2 + m(t− 2)(t− 1)2 + 2m2, DWM2(G) = (t(m + 1) − 1)((t− 3)(t(m + 1) − 1) + m(t2 − 3t + 4)) + m(t− 3)(t− 1)2, DWF(G) = (t− 2)(t(m + 1) − 1)3 + m(t− 2)(t− 1)3 + 2m3. Proof. Let G ∼= Bm,t be a stacked book graph with t(m + 1) vertices, where m ≥ 2 and t ≥ 3. There are t−2 vertices of downhill degree t(m + 1)−1, m(t−2) vertices of downhill degree t−1, 2 vertices of downhill degree m and 2m vertices of downhill degree 0. Hence, DWM1(G) = (t− 2)(t(m + 1) − 1)2 + m(t− 2)(t− 1)2 + 2m2, DWF(G) = (t− 2)(t(m + 1) − 1)3 + m(t− 2)(t− 1)3 + 2m3. Int. J. Anal. Appl. 19 (2) (2021) 216 Suppose that Ea,b = {uv ∈ E(G) : ddn(u) = a and ddn(v) = b}. The stacked book graph contains 6 types of edges E0,t−1,E0,m,Et(m+1)−1,m,Et(m+1)−1,t(m+1)−1,Et−1,t−1 and Et−1,t(m+1)−1 edges. In the Figure 2, the types of edges, E0,t−1,E0,m,Et(m+1)−1,m, Et(m+1)−1,t(m+1)−1,Et−1,t−1 and Et−1,t(m+1)−1 are colored in red, blue, green, yellow, pink and black, respectively. Figure 2. Stacked book graph Bm,t. Table 1 gives the number of edges in each type. Type Number of edges E0,t−1 2m E0,m 2m Em,t(m+1)−1 2 Et(m+1)−1,t(m+1)−1 t− 3 Et−1,t−1 m(t− 3) Et−1,t(m+1)−1 m(t− 2) Table 1. The number of edges in the different types of edges of stacked book graph. Thus, we get DWM2(G) = m(t− 3)(t− 1)2 + (t− 3)(t(m + 1) − 1)2 + m(t− 2)(t− 1)(t(m + 1) − 1) + 2m(t(m + 1) − 1) = (t(m + 1) − 1)((t− 3) + m(t− 2)(t− 1) + 2m) + m(t− 3)(t− 1)2. Int. J. Anal. Appl. 19 (2) (2021) 217 Hence, DWM2(G) = (t(m + 1) − 1)((t− 3)(t(m + 1) − 1) + m(t2 − 3t + 4)) + m(t− 3)(t− 1)2. � Proposition 3.2. Let G ∼= Fa,b,c be the firefly graph with 2a + 2b + c + 1 vertices. Then, DWM1(G) = (2a + 2b + c) 2 + 2a + b, DWM2(G) = (2a + 2b + c)(2a + b) + a, DWF(G) = (2a + 2b + c)3 + 2a + b. Proof. Let G ∼= Fa,b,c be the firefly graph with 2a + 2b + c + 1 vertices. It has one vertex of downhill degree 2a + 2b + c, 2a + b vertices of downhill degree 1 and b + c vertices of downhill degree 0. Hence, DWM1(G) = (2a + 2b + c) 2 + 2a + b, DWF(G) = (2a + 2b + c)3 + 2a + b. The firefly graph contains 4 types of edges E0,1,E1,1,E2a+2b+c,0 and E2a+2b+c,1 edges. In the Figure 3, the different types of edges E0,1,E1,1,E2a+2b+c,0 and E2a+2b+c,1 are colored in blue, red, green and black, respectively. Figure 3. Firefly graph Fa,b,c. Table 2, gives the number of edges in each type. Int. J. Anal. Appl. 19 (2) (2021) 218 Type Number of edges E0,1 b E1,1 a E0,2a+2b+c c E1,2a+2b+c 2a + b Table 2. The number of edges in the different types of edges of firefly graph. Hence, DWM2(G) = (2a + 2b + c)(2a + b) + a. � 4. The Downhill Zagreb Indices of Honeycomb Network and Graphene In this section, we obtain exact values for first, second and forgotten downhill indices for honeycomb network and Graphene. Honeycomb Network The honeycomb network very much important in computer graphics, cellular base stations, image processing and representation of benzene hydrocarbons in chemistry. The recursive use hexagonal tiling in a particular pattern, honeycomb networks are formed. Definition 4.1. The honeycomb network HC(1) is a hexagon and HC(2) is obtain by dding 6 hexagons to the boundary edges of HC(1). The honeycomb network HC(n) is obtained from HC(n− 1) by adding a layer of hexagons around the boundary of HC(n− 1). The number of vertices and edges of HC(n) are 6n2 and 9n2 − 3n respectively. Int. J. Anal. Appl. 19 (2) (2021) 219 Figure 4. Honeycomb network HC(4) with downhill degree of the vertices. Theorem 4.1. Let G ∼= HC(n) be a honeycomb network of dimension n, where n ≥ 3. Then, DWM1(G) = (6n 2 − 6n)(6n2 − 1)2 + 12, DWM2(G) = (9n 2 − 15n + 6)(6n2 − 1)2 + 72n2 − 6, DWF(G) = (6n2 − 6n)(6n2 − 1)3 + 12. Proof. Let G ∼= HC(n) be a honeycomb network of dimension n, where n ≥ 3. There are 2n lines in G as in Figure 4. By labeling the lines from up to down L1,L2, ...,L2n, it is clear to see that, L1 symmetric with L2n. L2 is symmetric with L2n−1 ... Ln is symmetric with Ln+1. The first line L1 has 2n + 1 vertices in which 4 vertices are of downhill degree 1, n− 2 vertices of downhill degree 0 and n− 1 vertices of downhill degree 6n2 − 1. The line Ln has 4n− 1 vertices in which 2 vertices of downhill degree 1 and 4n− 3 vertices of downhill degree 6nn − 1. Any line between L1 and Ln has two vertices of downhill degree 0 and the others of downhill degree 6n2 − 1. The number of vertices of downhill degree 1 is 12, the number of vertices of downhill degree 0 is 6(n− 2). Therefore, the number of vertices of downhill degree 6n2 − 1 is 6n2 − 6n. Hence, DWM1(G) = (6n 2 − 6n)(6n2 − 1)2 + 12, DWF(G) = (6n2 − 6n)(6n2 − 1)3 + 12. Int. J. Anal. Appl. 19 (2) (2021) 220 In a honeycomb network there are four types of edges based on the downhill degree of the vertices of each edge. The following table gives the four types and gives the number of edges in each type. Type Number of edges E1,1 6 E1,6n2−1 12 E0,6n2−1 12(n− 2) E6n2−1,6n2−1 9n 2 − 15n + 6 Thus, we get DWM2(G) = ∑ uv∈E(G) ddn(u)ddn(v) = |E1,1|(1)(1) + |E1,6n2−1|(1)(6n2 − 1) + |E0,6n2−1|(0)(6n2 − 1) + |E6n2−1,6n2−1|(6n2 − 1)(6n2 − 1) = 6 + 12(6n2 − 1) + (9n2 − 15n + 6)(6n2 − 1)2 = (9n2 − 15n + 6)(6n2 − 1)2 + 72n2 − 12 + 6. Hence, DWM2(G) = (9n 2 − 15n + 6)(6n2 − 1)2 + 72n2 − 6. � Graphene Graphene is a single layer of carbon atoms which are tightly bound in a hexagonal honeycomb lattice. Graphene is 200 times stronger than steel, one million times thinner than a human hair and word’s most conductive material and such properties attracted researchers and scientists to study more about Graphene. Graphene has unique properties which unlocks various applications from electronics to optics, sensors, and bio-devices. Theorem 4.2. Let G ∼= Gt,s be the graph of Graphene with t rows of benzene rings and s benzene rings in each row. The first downhill Zagreb index is given by DWM1(G) =   150 if t = 1,s = 1; 2(t− 1)(4t + 1)2 + 2(t + 34) if t > 1,s = 1; 2(s− 1)(4s + 1)2 + 72 if t = 1,s > 1; 2(st− 1)(2st + 2s + 2t− 1)2 + 2(t + 12) if t > 1,s > 1. Int. J. Anal. Appl. 19 (2) (2021) 221 Proof. Let G ∼= Gt,s be the graph of Graphene with t rows of benzene rings and s benzene rings in each row. Let v1,v2, ...,v2s+1 be the vertices of the first line L1. There are t + 1 lines in which two lines with 2s + 1 vertices and the others lines have 2s + 2 vertices as in Figure 5. We have four cases: Figure 5. Graphene Gt,s. Case 1. If t = 1 and s = 1. The graph G become cycle with 6 vertices and by Corollary 2.1, DWM1(G) = 6(5)2 = 150. Case 2. If t > 1 and s = 1.The number of vertices in this case is 4t + 2. The first line L1 has 3 vertices of downhill degree 3. By symmetry, it can be seen that the line Lt+1 has the same vertices as L1. The second line L2 has one vertex of downhill degree 3, one vertex of downhill degree 1 and 2 vertices of downhill degree 4t + 1. By symmetry, the line Lt has the same vertices as L2. Also, there are Lk lines, where k = t − 3, having 2 vertices of downhill degree 1 and 2 vertices of downhill degree 4t + 1. Thus, we get DWM1(G) = 2(t− 1)(4t + 1)2 + 2(t + 34). Case 3. If t = 1 and s > 1. In this case, there are only two lines and each line has 2s + 1 vertices. The first line L1 has 4 vertices of downhill degree 3, s− 2 vertices of downhill degree 0 and s− 1 vertices of downhill degree 4s + 1. By symmetry, it can be seen that the line L2 has the same vertices as L1. Thus, we get DWM1(G) = 2(s− 1)(4s + 1)2 + 72. Case 4. If t > 1 and s > 1. The number of vertices in this case is 2st + 2s + 2t. Let n = 2st + 2s + 2t. The first line L1 has two vertices of downhill degree 2, two vertices of downhill degree 1, s−2 vertices of downhill degree 0 and s−1 vertices of downhill degree n−1. By symmetry, it can be seen that the line Lt+1 has the Int. J. Anal. Appl. 19 (2) (2021) 222 same vertices as L1. The second line L2 has one vertex of downhill degree 2, one vertex of downhill degree 1 and 2s vertices of downhill degree n− 1. By symmetry, the line Lt has the same vertices as L2. Also, there are k = t− 3 lines, each line have 2 vertices of downhill degree 1 and 2s vertices of downhill degree n− 1. Thus, the first downhill Zagreb index of a Graphene is given by, DWM1(G) = 2(st− 1)(2st + 2s + 2t− 1)2 + 2(t + 12). � Theorem 4.3. Let G ∼= Gt,s be a graph of Graphene with t rows of benzene rings and s benzene rings in each row. The second downhill Zagreb index is given by DWM2(G) =   150 if t = 1,s = 1; (4t + 1)(8t2 − 8t + 5) + t + 52 if t > 1,s = 1; (4s + 1)(4s2 − 3s + 11) + 54 if t = 1,s > 1; α if t > 1,s > 1, where α = (2st + 2s + 2t− 1) ((s(3t− 2) − (1 + t))(2st + 2s + 2t− 1) + 2(t + 4)) + t + 16. Proof. Let G ∼= Gt,s be the graph of Graphene with t rows of benzene rings and s benzene rings in each row. So the graph has 2st + 2s + 2t vertices and 3st + 2s + 2t− 1 edges. We have four cases: Case 1. If t = 1 and s = 1. The graph G become cycle with 6 vertices and by Corollary 2.1, DWM2(G) = 6(5)2 = 150. Case 2. If t > 1 and s = 1. The number of vertices in this case is 4t + 2 and the number of edges is 5t + 1. Two dimensional structure of Graphene contains the following types of edges E1,1,E3,3,E1,4t+1,E3,4t+1 and E4t+1,4t+1. In Figure 6 the edge types, E1,1,E3,3,E1,4t+1,E3,4t+1 and E4t+1,4t+1 are colored in red, blue, green, yellow and black, respectively. Int. J. Anal. Appl. 19 (2) (2021) 223 Figure 6. Graphene Gt,s with t > 1,s = 1. The number of edges in E1,1,E3,3,E1,4t+1,E3,4t+1 and E4t+1,4t+1 in each row is mention in the following table. Row |E1,1| |E3,3| |E1,4t+1| |E3,4t+1| |E4t+1,4t+1| 1 0 3 1 2 1 2 1 0 2 0 2 3 1 0 2 0 2 4 1 0 2 0 2 . . . . . . . . . . . . . . . . . . t− 1 1 0 1 1 2 t 0 3 0 1 0 Total t− 2 6 2 + 2(t− 3) 4 1 + 2(t− 2) Therefore, we have |E1,1| = t− 2 edges, |E3,3| = 6 edges, |E1,4t+1| = 2t− 4 edges, |E3,4t+1| = 4 edges and |E4t+1,4t+1| = 2t− 3 edges. Int. J. Anal. Appl. 19 (2) (2021) 224 DWM2(G) = ∑ uv∈E(G) ddn(u)ddn(v) = |E1,1|(1)(1) + |E3,3|(3)(3) + |E1,4t+1|(1)(4t + 1) + |E3,4t+1|(3)(4t + 1) + |E4t+1,4t+1|(4t + 1)(4t + 1) = t− 2 + 6 × 9 + (2t− 4)(4t + 1) + 4 × 3(4t + 1) + (2t− 3)(4t + 1)(4t + 1) = (4t + 1)(2t− 4 + 12 + 8t2 + 2t− 12t− 3) + t + 52. Hence, DWM2(G) = (4t + 1)(8t 2 − 8t + 5) + t + 52. Case 3. If t = 1 and s > 1. The number of vertices in this case is 4s + 2 and the number of edges is 5s + 1. Two dimensional structure of Graphene, we have E3,3,E3,4s+1,E0,4s+1 and E4s+1,4s+1 edges. In Figure 7, we have colored E3,3,E3,4s+1,E0,4s+1 and E4s+1,4s+1 edges in red, blue, green and black, respectively. Figure 7. Graphene Gt,s with t = 1,s > 1. Therefore, we have |E3,3| = 6 edges, |E3,4s+1| = 4 edges, |E0,4s+1| = 4(s−2) edges and |E4s+1,4s+1| = s−1 edges. DWM2(G) = ∑ uv∈E(G) ddn(u)ddn(v) = |E3,3|(3)(3) + |E3,4s+1|(3)(4s + 1) + |E0,4s+1|(0)(4s + 1) + |E4s+1,4s+1|(4s + 1)(4s + 1) = 6 × 9 + 4 × 3(4s + 1) + 4(s− 2)(0)(4s + 1) + (s− 1)(4s + 1)(4s + 1) = (4s + 1)(12 + 4s2 + s− 4s− 1) + 54. Hence, DWM2(G) = (4s + 1)(4s 2 − 3s + 11) + 54. Case 4. If t > 1 and s > 1. Let n = 2st + 2s + 2t. Two dimensional structure of Graphene con- tains the following types of edges E1,1,E2,2,E0,n−1,E1,n−1,E2,n−1 and En−1,n−1. In Figure 8, the edges Int. J. Anal. Appl. 19 (2) (2021) 225 E1,1,E2,2,E0,n−1,E1,n−1,E2,n−1 and En−1,n−1 are colored in red, blue, green, yellow, pink and black, re- spectively. Figure 8. Graphene Gt,s with t > 1,s > 1. The number of edges in E1,1,E2,2,E0,n−1,E1,n−1,E2,n−1 and En−1,n−1 in each row is mention in the following table. Row |E1,1| |E2,2| |E0,n−1| |E1,n−1| |E2,n−1| |En−1,n−1| 1 1 2 2(s− 2) 3 2 3s− 2 2 1 0 0 2 0 3s− 1 3 1 0 0 2 0 3s− 1 . . . . . . . . . . . . . . . . . . . . . t− 1 1 0 0 1 1 3s− 1 t 1 2 2(s− 2) 2 1 s− 1 Total t 4 4(s− 2) 2t 4 s(3t− 2) − (1 + t) Therefore, we have |E1,1| = t edges, |E2,2| = 4 edges, |E0,n−1| = 4(s − 2) edges, |E1,n−1| = 2t edges, |E2,n−1| = 4 edges and |En−1,n−1| = s(3t− 2) − (1 + t)edges. Int. J. Anal. Appl. 19 (2) (2021) 226 DWM2(G) = ∑ uv∈E(G) ddn(u)ddn(v) = |E1,1|(1)(1) + |E2,2|(2)(2) + |E0,n−1|(0)(n− 1) + |E1,n−1|(1)(n− 1) + |E2,n−1|(2)(n− 1) + |En−1,n−1|(n− 1)(n− 1) = t + 4 × 9 + 2t(n− 1) + 4 × 2(n− 1) + s(3t− 2) − (1 + t))(n− 1)(n− 1) = (n− 1)(2t + 8 + (s(3t− 2) − (1 + t))(n− 1)) + t + 16. Hence, DWM2(G) = (2st + 2s + 2t− 1)((s(3t− 2) − (1 + t))(2st + 2s + 2t− 1) + 2(t + 4)) + t + 16. � Proposition 4.1. Let G ∼= Gt,s be the graph of Graphene with t rows of benzene rings and s benzene rings in each row. The forgotten downhill Zagreb index is given by DWF(G) =   750 if t = 1,s = 1; 2(t− 1)(4t + 1)3 + 2(t + 106) if t > 1,s = 1; 2(s− 1)(4s + 1)3 + 216 if t = 1,s > 1; 2(st− 1)(2st + 2s + 2t− 1)3 + 2(t + 24) if t > 1,s > 1. Proof. The proof similar to the proof Theorem 4.2. � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] A. Alwardi, A. Alqesmah, R. Rangarajan, I.N. Cangul, Entire Zagreb indices of graphs, Discrete Math. Algorithm. Appl. 10 (2018), 1850037. [2] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, Berlin, Germany, 2008. [3] J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of molecular descriptors: the equivalence of Zagreb indices and walk counts, MATCH Commun. Math. Comput. Chem. 54 (2005), 163-176. [4] B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), 1184–1190. [5] M. Ghorbani, M.A. Hosseinzadeh, The third version of zagreb index, Discrete Math. Algorithm. Appl. 05 (2013), 1350039. [6] I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92. [7] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals, Total π−electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535-538. [8] J. Deering, Uphill & Downhill Domination in Graphs and Related Graph Parameters. Thesis, East Tennessee State Uni- versity, 2013. Int. J. Anal. Appl. 19 (2) (2021) 227 [9] F. Harary, Graph theory, Addison-Wesley, Reading Mass, 1969. [10] S.T. Hedetniemi, T.W. Haynes, J.D. Jamieson, W.B. Jamieson, Downhill domination in graphs, Discuss. Math., Graph Theory. 34 (2014), 603–612. [11] M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009), 804-811. [12] J. Kok, N.K. Sudev, U. Mary, On chromatic Zagreb indices of certain graphs, Discrete Math. Algorithm. Appl. 09 (2017), 1750014. [13] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after, Croatica Chemica Acta, 76 (2003), 113-124. [14] A. Saleh, A. Aqeel, I. N. Cangul, On the entire ABC index of graphs, Proc. Jangjeon Math. Soc. 23 (2020), 39-51. [15] Anwar Saleh, Najat Muthana, Wafa Al-Shammakh, Hanaa Alashwali, Monotone chromatic number of graphs, Int. J. Anal. Appl. 18 (2020), 1108-1122. [16] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004), 113-118. [17] B. Zhou, I. Gutman, Relations between Wiener, hyper-Wiener and Zagreb indices, Chem. Phys. Lett. 394 (2004), 93-95. [18] B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54 (2005), 233-239. [19] S. Wang, B. Wei, Multiplicative Zagreb Indices of Cacti, Discrete Math. Algorithm. Appl. 8 (2016), 1650040. 1. Introduction 2. Downhill Zagreb Indices 3. The Downhill Zagreb Indices of Graphs under Some Binary Operations 4. The Downhill Zagreb Indices of Honeycomb Network and Graphene References