On Lanzhou Index of Graphs

: Let G = ( V , E ) be a simple graph with vertex set V ( G ) and edge set E ( G ). The Lanzhou index of a graph G is defined by Lz ( G ) = (cid:80) u ∈ V ( G ) d 2 u d u , where d u ( d u resp.) denotes the degree of the vertex u in G ( G , the complement graph of G resp.). It has predictive powers to provide insights of chemical relevant properties of chemical graph structures. In this paper we discuss some properties of Lanzhou index. Several inequalities having lower and upper bound for the Lanzhou index in terms of first, second and third Zagreb indices, radius of graph, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index, are discussed. At the end the Lanzhou index of corona and join of graphs have been derived.


Introduction and Motivation
Throughout the paper we consider simple, undirected, unweighted graphs only unless it is specified.Let G = (V, E) be a graph with vertex set V(G) and edge set E(G).The number of vertices and edges are denoted by |V(G)| and |E(G)|, respectively.The degree of a vertex u ∈ V(G), denoted by d G (u) (simply d u whenever understood) is the number of adjacent vertices to u in G.The distance between any two vertices u and v, denoted by d(u, v) is defined as the length of shortest path between u and v in G.The complement graph G of a graph G is the graph with the same vertex set V(G) and the vertices are adjacent in G if and only if they are not adjacent in G.
The Zagreb indices were first introduced by Gutman and Trinajstić [1], they are important molecular descriptors and have been closely correlated with many chemical properties [2].The first Zagreb index M 1 (G) of a graph G is defined as while the second Zagreb index M 2 (G) is defined as Furtula and Gutman [3] introduced forgotten topological index (also called F-index) which is defined as In [4], Vukičević et al. considered a linear combination of M 1 (G) and F(G) of the form M 1 (G) + λF(G), where λ was a free parameter ranging from −20 to 20.From the above linear combination, Vukičević et  where d u is the degree of the vertex u in G.For its mathematical properties see the paper [4].For self complimentary graphs, d u = d u , implying that the Lanzhou index is same as forgotten topological index.In chemical graph theory, many vertex degree based topological indices and their properties have been investigated in [5][6][7][8][9][10][11][12][13][14][15][16][17][18].
In this paper we first discuss some properties of Lanzhou index in Section 2. An upper bound of Lanzhou index for unicyclic graphs has been obtained.The relationships between Lanzhou index and other topological indices such as graph radius, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index are derived in Section 3. At the end in Section 4 the Lanzhou index of the join and corona of graphs are provided.

Some properties of Lanzhou index
In this section we discuss the properties of the Lanzhou index.
Proof.We have

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From the definition of Lanzhou index it is clear that the value of Lz(G) is a positive integer.The next result shows that the Lanzhou index of a graph is a positive even integer.
Theorem 1.For any graph G, the Lanzhou index Lz(G) is even.
Proof.Let G be a graph with n vertices.By definition of Lanzhou index we have It is clear to see that the first term of the sum in (1) is even.For the second term of the sum in (1) we have the following two cases: On Lanzhou Index of Graphs 145 Case 1 : n is even.Since d u is odd, we have d u = n − 1 − d u is even.And in this case d 2 u d u is even.Therefore the second term of the sum in (1) is even.
Case 2 : n is odd.Since d u is odd, we have d u = n − 1 − d u is odd.And in this case d 2 u d u is odd.It is well known that the number of odd degree vertices in a graph is even.Using the fact that the sum of even number of odd numbers is even, it implies that the second term of the sum in (1) is even.
Hence, for any graph, the Lanzhou index Lz(G) is even.□ The union of two graphs G and H denoted by G ∪ H is the graph with vertex set V(G) ∪ V(H) and edge set E(G) ∪ E(H).We know that complete graphs and null graphs (graphs with isolated vertices) are the only example of graphs with minimum Lanzhou index 0. Likewise, the path of length 2 and K 2 ∪ K 1 are the graphs with second minimum Lanzhou index 2.
Proposition 2. For any graph G, Lz(G) = 2 if and only if G P 3 or G K 2 ∪ K 1 . Proof.
Otherwise, we have n ≥ 4 and there exist two vertices v and This completes the proof.□ The bound for Lanzhou index of any graph is provided and the graph with extremal value have been characterized in [4].

Proposition 3. [4]
Let G be any graph with n vertices.Then In the following we give some lower and upper bounds for any graph with n vertices and m edges having minimum degree δ and maximum degree ∆.
Theorem 2. Let G be a connected graph with n vertices and m edges having minimum degree δ and maximum degree ∆.Then with both equalities hold if and only if G is a regular graph.
Proof.Since δ ≤ d u ≤ ∆, by definition of Lanzhou index, we have From the above result, we get the lower bound.Moreover, the equality holds in the lower bound if and only if where M 1 (G) is the first Zagreb index of graph G.
Proof.We have Theorem 4. Let G be a graph with n vertices and e = uv be an edge in G.
Proof.By the definition of Lanzhou index, we have Therefore we have This completes the proof of the theorem.□ Let T n be a tree with n vertices and T ∆ n denotes the set of all trees on n vertices with maximum degree at most ∆.Vukičević et al. [4] obtained the following result: Proposition 4. [4] Let n ≥ 8 be an integer and T n ∈ T 4 n .Then Theorem 5. Let G be a unicyclic graph with n (≥ 8) vertices and On Lanzhou Index of Graphs 147 Proof.Let e = uv be an edge in G such that H = G − e, where H is a tree of order n.Then by Theorem 4, we see that where d u and d v are the degrees of u and v, respectively.Since d u , d v ≤ 4, from the above result, we have Since H is a tree, by Proposition 4, we obtain This completes the proof of the theorem.□

Relationships between Lanzhou index and other topological indices
The eccentricity of a vertex v in a graph G is defined as

Moreover, the above equality holds together for all vertices in G if and only if
, where K n − iK 2 denotes the graph obtained by removing i independent edges from K n .We now give a relation between Lz and M 1 .
Moreover, the equality holds in (2) if and . Now by the definition of Lanzhou index, we have Suppose that equality holds in (2).Then we have n Proof.From the definition of Lanzhou index with the given condition, we have

□
We now mention two more relations between Lz, M 1 and M 2 .
Theorem 8. Let G be a graph of order n.Then with equality holding if and only if each connected component of G is regular.Moreover, with both equalities hold if and only if G is a regular graph.
Proof.One can easily see that F(G) ≥ 2M 2 (G) with equality holding if and only if each connected component of G is regular, and δ M 1 (G) ≤ F(G) ≤ ∆ M 1 (G) with both equalities hold if and only if G is a regular graph.Since Lz(G) = (n − 1) M 1 (G) − F(G), using the above results, we get the required results.This completes the proof of the theorem.□ The eccentric connectivity index [18] of a graph G, denoted by ξ c (G) is defined as Here we give a relation between Lz, M 1 and ξ c .
Theorem 9. Let G be a graph of order n and minimum degree δ.Then with equality holding if and only if G K n or G K n − n 2 K 2 (n is even).Proof.From the definition of Lanzhou index, we have By the proof of the Theorem 6, one can easily see that the equality holds if and only if G K n or G K n − n 2 K 2 (n is even).□ We now give a relation between Lz and M 2 .
Theorem 10.Let G be a connected graph with n vertices and minimum degree δ.Then with both equalities hold if and only if G is a regular graph.
Proof.We construct an auxiliary real valued function of two variables x and y as Therefore, g(x, y) is monotonically decreasing in the variable x.Since the function g(x, y) is symmetric in both x and y, it is also monotonically decreasing in the variable y.Thus we have g(x, y) attains its maximum value at (δ, δ) and the minimum value at (∆, ∆).Hence This implies that Using the above results with the definition of the Lanzhou index, we have and Moreover, both equalities hold in (3) if and only if Since G is connected, both equalities hold in (3) if and only if G is a regular graph.□ The irregularity of a graph G, denoted by irr(G) is defined as It is also called third Zagreb index of graph.More results on irregularity, one can find in [20][21][22].
Here we give a relation between Lz with M 1 (G), M 2 (G), irr(G) and F(G) of graph G.
Theorem 11.Let G be a connected graph with n vertices.Then with left (right) equality holding if and only if G is a regular graph (G is a regular graph or a bipartite semiregular graph).
Proof.From the definition of irregularity, we have as Lz(G) = (n − 1)M 1 (G) − F(G).Hence we get the right inequality.Moreover, the equality holds if and only if |d u − d v | = ∆ − δ for all uv ∈ E(G), that is, if and only if G is a regular graph or a bipartite semiregular graph as G is connected.
Hence we get the left inequality.Moreover, the left equality holds if and only if d u = d v for all edges uv ∈ E(G), that is, if and only if G is a regular graph as G is connected.□ The Schultz index of a molecular graph G, introduced by Schultz [23], is defined as On Lanzhou Index of Graphs 151 where d(u, v) denotes the distance between the vertices u and v.The Schultz indices have been shown as useful descriptor for molecular design and characterization with desired properties in [2,24].
The join G ∨ H of two simple graphs G and H is the graph with the vertex set V(G ∨ H) = V(G)∪V(H) and the edge set or G is a regular graph with diameter 2.
Proof.By definition of Schultz index we have The first part of the proof is done.
Suppose that equality holds.Then d(u, v) = 1 or 2 for any pair of vertices (u, v).Moreover, d u = d v = δ when d(u, v) = 2 for any pair of vertices (u, v), that is, all the vertices in G have degree either n − 1 or δ.
Then all the vertices in G are of degree δ.Hence G is a regular graph with diameter 2.
Let G be a regular graph with diameter 2. Then This completes the proof of the theorem.□ The inverse sum indeg (ISI) index [16] is used as a significant predictor of total surface area for octane isomers.The ISI index is defined as Theorem 13.Let G be a graph with n vertices and maximum degree ∆.Then with equality holding if and only if G is a regular graph.
Proof.Since ∆ is the maximum degree in G, we have with equality holding if and only if One can easily see that with equality holding if and only if d u = d v .
Using the above results with the definition of the Lanzhou index, we have Proof.Since IS I(G) ≥ m 2 n , from Theorem 13, we obtain the required result.Moreover, the equality holds if and only if G is a regular graph.□ The symmetric division deg index, S DD, was defined in [17] as For recent results on S DD(G) see the papers [25][26][27][28][29] and the references cited therein.Here we give a relation between Lz and S DD.The first part of the proof is done.
The equality holds if and only if d u = d v = ∆ for any edge uv ∈ E(H) and p = 0, that is, if and only if G is a regular graph.□ al. introduced in the same paper a new topological index named as Lanzhou index.It is denoted by Lz(G) and defined by Lz( H) .The following theorem gives a relation between Schultz index and Lanzhou index.Theorem 12. Let G be a connected graph of order n.Then S I(G) ≥ 1 2(n − 1) Lz(G) + F(G) + n(n − 1) − 2m δ with equality holding if and only

Corollary 2 .
n − 1 − ∆)IS I(G).Moreover, the equality holds if and only if d u = d v = ∆ for any edge uv ∈ E(G), that is, if and only if G is a regular graph.□Let G be a graph with n vertices, m edges and maximum degree ∆.ThenLz(G) ≥ 4m 2 (n − 1 − ∆) nwith equality holding if and only if G is a regular graph.
n is even).Conversely, one can easily see that the equality holds in (2) for K n or for K n − n 2 K Let G be a graph of order n.If d u + d v ≥ n for each edge uv ∈ E(G), then Lz 2 (n is even).□Herewe give a relation between Lz, M 1 and M 2 .