On the Uphill Zagreb Indices of Graphs

One of the tools, to research and investigation the structural dependence of various properties and some activities of chemical structures and networks is the topological indices of graphs. In this research work, we introduce novel indices of graphs which they based on the uphill degree of the vertices termed as uphill Zagreb topological indices. Exact formulae of these new indices for some important and famous families of graphs are established.


Introduction
In this research, by graphs, we mean undirected finite simple graph. We denote G = (V, E) for a graph, where V is the set of vertices and E is the set of edges. For a vertex v ∈ V (G) the degree of v , d(v ) is the number of edges incident with v . Any terminology or notation which, we did not mention its definition, we refer the reader to [3].
Topological indices have a widespread position specifically in pharmacology, chemistry, networks and many others. (see [8, 9, 13-15, 18, 24, 25]). Almost of the indices of contemporary interesting in mathematical chemistry are introduced based on vertex degrees of the chemical graph. The two well-known topological indices of graphs are the Zagreb indices that have been introduced by Gutman and Trinajstic by their work in [16], and described as M 1 (G) = u∈V (G) (d (u)) 2 and M 2 (G) = uv ∈E(G) d (u) d (v ), respectively. The forgotten topological index was introduced by Furtula and Gutman [10] as F (G) = u∈V (G) (d (u)) 3 .
Zagreb indices were studied considerably due to their numerous applications inside the area of present chemical methods which want extra time and more charges. Many new reformulated and prolonged versions of the Zagreb indices have been delivered for several similar reasons (cf. [1,4,12,19,22,[27][28][29]).
The uphill domination and some related concepts are introduced and studied in ( [7,23]). In this paper motivated by the large applications of topological indices and the concept of uphill domination, we introduce novel indices of graphs based on a new degree (uphill degree) of the vertices termed as uphill Zagreb topological indices. Some properties and exact formulae of these new topological indices for some standard and famous families of graphs are established.

Some Results on the Uphill Zagreb Indices of Graphs
Definition 2.1. [7] For any graph G = (V, E). A path u − v is a sequence of vertices in G, initialing with u and terminal at v , such that sequential vertices are adjacent, and no vertex is repeated. A path For any vertices u and v in G, if there is an uphill path from u to v we say that u is uphill adjacent An uphill neighborhood of the vertex v is denoted by N up (v ) and described as: The maximum and minimum uphill degrees in the graph G denoted by ∆ up (G) and δ up (G), respectively.
The vertex with uphill degree equal to zero is called uphill isolated vertex.
In this paper by E x,y , we mean that E x,y = {uv ∈ E(G) : d up (u) = x and d up (v ) = y }. Definition 2.3. For any graph G = (V, E) the first uphill Zagreb, second uphill Zagreb, forgotten uphill Zagreb index and modified first uphill Zagreb are defined as: Lemma 2.1. Suppose G be a graph, for any two vertices u and v , where u is uphill adjacent to v , if Proof. If G is a graph, where u and v be any two vertices in G, where u is uphill adjacent to v . Then there are two cases: where r = d up (v ). In this case, Hence, Case 2. When u not adjacent with v . Then, there is an uphill path Γ from u to v . Clearly, Hence, Theorem 2.1. Let G be any graph. Then, the graph G is regular if and only if G is uphill regular.
Proof. If G is regular graph of n vertices. Then it is straightforward, that G is n − 1 uphill regular graph.
To prove the other direction of the theorem, we will prove that if G is not regular then G is not uphill regular. Suppose that G is not regular. Then, there exist at least two vertices say u and v such that . We have two cases: If u is uphill adjacent to v . Then, By Lemma 2.1, the graph G is not uphill regular.   Proof. Let G be any graph and let τ (G) be the number of uphill paths in G, for any vertex v in G it is obviously from the definition of uphill degree of the vertex that, d up (v ) is less than or equal the number of paths in G which originated from v . Also if we denote by Γ v as the number of uphill paths Furthermore, it is easy to check that if G is acyclic, then d up (v ) = Γ v . Hence the equality holds.
Proposition 2.3. Let G ∼ = P n , be a path of n ≥ 3 vertices. Then, Proof. Suppose G ∼ = P n be a path of n ≥ 3 vertices. Then there are two vertices with uphill degree n − 2 and n − 2 vertices of uphill degree n − 3. So by using the definition of first uphill index we get, Similarly, There are two edges of the type E n−2,n−3 and n − 3 edges of the type E n−3,n−3 . Then, In the same way we get be a regular graph of degree k and has n vertices. Then, Proof. Let G = (V, E) be k− regular graph with n vertices. Then it isn't difficult to see that between any two vertices of G there exists an uphill path, so for any v ∈ V (G), we have d up (v ) = n −1. Hence, Corollary 2.1. For any complete graph K n , we have By a graph W n , we mean a wheel graph of n + 1 vertices.
Proof. Let G ∼ = W n , where n ≥ 3 be a wheel graph. The graph G has one vertex of uphill degree zero and n vertices of uphill degree n. Then, In the same way, we get the forgotten uphill index There are n edges of the type E n,n and n edges of the type E n,0 . Then, A tadpole graph T m,n is constructed by joining between C m and P n by a bridge [17].
Proposition 2.6. Let G ∼ = T m,n , where m, n ≥ 3 be a tadpole graph of m + n vertices. Then, Proof. Let G ∼ = T m,n , where m, n ≥ 3 be a tadpole graph of m + n vertices. There are m − 1 vertices of uphill degree m − 1, one vertex of uphill degree zero, n − 1 vertices of uphill degree n − 1 and one vertex of uphill degree n. Then, we get Similarly,

Type
Number of edges Table 1. Edge partition of tadpole graph based on uphill degree of end vertices. Now, by using the partition given in Table1, we get Also, The graph which obtained from a wheel graph with extra vertex between each pair of adjacent vertices of the outer cycle is called gear graph G n [17].
Proposition 2.7. Let G ∼ = G n , where n ≥ 4 be a gear graph. Then, Proof. Let G ∼ = G n , where n ≥ 4 be a gear graph. Then, there are n vertices of uphill degree one, one vertex of uphill degree zero and n vertices of uphill degree three. Clearly, we get In the same way, we get There are n edges of the type E 1,0 and 2n edges of the type E 1,3 . Then, The windmill graph W d(s, k), where s, k ≥ 2, is a graph of k copies of complete graph K s at a shared common vertex [17].
, where s ≥ 3 and k ≥ 2 be a windmill graph of k(s − 1) + 1 vertices. Then, Proof. Let G ∼ = W d(s, k), where s ≥ 3 and k ≥ 2 be a windmill graph of k(s − 1) + 1 vertices. The graph G has one vertex of uphill degree zero and k(s − 1) vertices of uphill degree s − 1. So, Similarly, There are sk(s−1) 2 − k(s − 1) edges of the type E s−1,s−1 and k(s − 1) edges of the type E s−1,0 . Then, Also, The graph which is obtained from W n by adding an end edge to each outer vertex of W n , is called helm graph and denoted by H n [17]. iii. UP M * 2 (G) = 5n 2 + n, iv. UP F (G) = 2n 4 + 3n 3 + 3n 2 + n.
Proof. Let G ∼ = H n , where n ≥ 3 be a helm graph. The graph G has only one vertex of uphill degree zero, n vertices of uphill degree n + 1 and n vertices of uphill degree n. Then, UP M 1 (G) = n(n + 1) 2 + n(n) 2 = 2n 3 + 2n 2 + n.
There are n edges of the type E n+1,n , n edges of the type E n,n and n edges of the type E n,0 . Then, Similarly, UP M * 1 (G) = n(2n + 1) + 2n(n) + n 2 = 5n 2 + n.
The double star graph S r,t which obtained from complete graph of two vertices by joining r pendent edges to one vertex and t pendent edges to the other vertex of the complete graph K 2 [17].
where r, t ≥ 2 be a double star graph. Then, Proof. Let G ∼ = S r,t , where r, t ≥ 2 be a double star with r + t + 2 vertices. Then, i. There are two cases: iii. Similarly as in part i i , if r = t, then UP M * 1 (G) = 6r + 2. If r < t, then UP M * 1 (G) = 3r + t + 1. iv. The proof is similarly to part i .
The subdivision graph S(G) of the graph G is a graph obtained from G by replacing each of its edge by a path of length 2. By simple calculation, we get the uphill Zagreb indices for the subdivision graphs of path, cycle and complete graph. For each p ≥ 0, the p-sun tree, denoted by Su p , is the tree of order n = 2p + 1 formed by taking the star on p + 1 vertices and subdividing each edge. For p, q ≥ 0, the (p, q)-double sun, denoted by DSu p,q , is the tree of order n = 2(p + q + 1) obtained by connecting the centers of DSu p and DSu q with an edge [11]. 13p + 5q + 1 i f p < q.
ii. UP M 2 (G) = The central graph of a graph G is obtained by subdividing each edge of G exactly once and joining all the non-adjacent vertices of G and denoted by C(G) [2].  Table 2. Edge partition of C(P n ) graph based on uphill degree of end vertices. Now, by using the partition in Table 2, we get UP M 2 (G) = 2n (n − 1) 2 + (n − 1) 3 (n − 2) 2 = n 4 − n 3 + n 2 − 3n + 2 2 .
By using the same partition in Table 2, we get ii. We have two cases: edges where each edge of the type E 1,0 , then UP M 2 (G) = 0.
iii. As part i i , we get UP M * 1 (G) = r s + 2r if r ≥ 3 and if r = 2, UP M * 1 (G) = r s − 2r + 24 . iv. In the same way as part i .
A firecracker graph F r,s is a graph obtained by the concatenation of n stars, each consists of s vertices, by linking one leaf from each star [6].
Theorem 2.4. Let G ∼ = F r,s , where s ≥ 5 be a firecracker graph with r s vertices. Then, ii. There are two cases: Case 1. If r = 2, the graph G has 2(s − 2) edges of type E 1,0 , two edges of the type E 0,3 and one edge of type E 3,3 . So, UP M 2 (G) = 9.
iv. As the method in i .
Book graph is a Cartesian product of a star and single edge, denoted by B r . The r -book graph is defined as the graph Cartesian product S r +1 × P 2 , where S r +1 is a star graph and P 2 is the path graph. The stacked book graph of order (r, t) is defined as the graph Cartesian product S r +1 × P t , where S r is a star graph and P t is the path graph on t nodes, and it is denoted by B r,t [17]. ii. UP M 2 (G) = 15r + 1, iii. UP M * 1 (G) = 14r + 2, iv. UP F (G) = 2(27r + 1).
The graph G has three kinds of edges, one edge of the type E 1,1 , 2r edges of the type E 3,1 and r edges of the type E 3,3 . Then, UP M 2 (G) = 15r + 1.
In Figure 1, we can see the graph G has 2r vertices are labeled by of uphill degree t − 3. Then, Similarly, There are 6 types of edges. Table 3. Edge partition of B r,t graph based on uphill degree of end vertices.
A firefly graph F a,b,c is a graph of n = 2a + 2b + c + 1 vertices that consists of c pendant edges, a triangles, and b pendant paths of length 2, all of them sharing a common vertex [5]. iii. UP M * 1 (G) = 8a + 4b + c, iv. UP F (G) = 16a + 9b + c.
Proof. Let G ∼ = F a,b,c be the firefly graph with 2a + 2b + c + 1 vertices. In Figure 2 Table 4. Edge partition of F a,b,c graph based on uphill degree of end vertices.
In Figure 2, the types of edges, E 2,2 , E 2,0 , E 1,0 and E 2,1 are colored by blue, red, black blue and green, respectively. Now, by using the edge partition in Table 4