On Lanzhou Index of Graphs

Bera, Somnath; Das, Kinkar

doi:10.61091/ars157-14

Abstract

References

Ars Combinatoria

Volume 157
Pages: 143-157

Research article

On Lanzhou Index of Graphs

^¹, ^²

¹School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China.

²Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea.

Received: 12/01/2019
Accepted: 05/05/2019
Published: 25/12/2023

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Abstract

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 $L z (G) = \sum_{u \in V (G)} d_{u}^{2} {\bar{d}}_{u}$ , where $d_{u}$ ( ${\bar{d}}_{u}$ resp.) denotes the degree of the vertex $u$ in $G$ ( $\bar{G}$ , 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.

Keywords: Lanzhou index, Zagreb indices, forgotten topological index, irregularity index, eccentric connectivity index, Schultz index

1. 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 $u \in V (G)$ , 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 $\bar{G}$ of a graph $G$ is the graph with the same vertex set $V (G)$ and the vertices are adjacent in $\bar{G}$ 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 $M_{1} (G) = \sum_{u \in V (G)} d_{u}^{2} = \sum_{u v \in E (G)} (d_{u} + d_{v}) .$ while the second Zagreb index $M_{2} (G)$ is defined as $M_{2} (G) = \sum_{u v \in E (G)} d_{u} d_{v} .$ Furtula and Gutman[3] introduced forgotten topological index (also called F-index) which is defined as $F (G) = \sum_{u \in V (G)} d_{u}^{3} = \sum_{u v \in E (G)} (d_{u}^{2} + d_{v}^{2}) .$

In [4], Vukičević et al. considered a linear combination of $M_{1} (G)$ and $F (G)$ of the form $M_{1} (G) + λ F (G)$ , where $λ$ was a free parameter ranging from $- 20$ to $20$ . From the above linear combination, Vukičević et al. introduced in the same paper a new topological index named as Lanzhou index. It is denoted by $L z (G)$ and defined by $L z (G) = \sum_{u \in V (G)} d_{u}^{2} {\bar{d}}_{u},$ where ${\bar{d}}_{u}$ is the degree of the vertex $u$ in $\bar{G}$ . For its mathematical properties see the paper[4]. For self complimentary graphs, $d_{u} = {\bar{d}}_{u}$ , 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.

2. Some properties of Lanzhou index

In this section we discuss the properties of the Lanzhou index.

Proposition 1. For a graph $G$ , $L z (G) = \sum_{u v \in E (G)} (d_{u} {\bar{d}}_{u} + d_{v} {\bar{d}}_{v})$ .

Proof. We have $\begin{array}{rcl} L z (G) & = & a \sum_{u \in V (G)} d_{u}^{2} (n - 1 - d_{u}) \\ = & \sum_{u v \in E (G)} (d_{u} (n - 1 - d_{u}) + d_{v} (n - 1 - d_{v})) \\ = & \sum_{u v \in E (G)} (d_{u} {\bar{d}}_{u} + d_{v} {\bar{d}}_{v}) . \end{array}$

From the definition of Lanzhou index it is clear that the value of $L z (G)$ 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 $L z (G)$ is even.

Proof. Let $G$ be a graph with $n$ vertices. By definition of Lanzhou index we have $\begin{matrix} (1) & L z (G) = \sum_{u \in V (G)} d_{u}^{2} {\bar{d}}_{u} = \sum_{\begin{matrix} u \in V (G), \\ d_{u} is even \end{matrix}} d_{u}^{2} {\bar{d}}_{u} + \sum_{\begin{matrix} u \in V (G), \\ d_{u} is odd \end{matrix}} d_{u}^{2} {\bar{d}}_{u} . \end{matrix}$

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:

$C a s e 1 :$ $n$ is even. Since $d_{u}$ is odd, we have ${\bar{d}}_{u} = n - 1 - d_{u}$ is even. And in this case $d_{u}^{2} {\bar{d}}_{u}$ is even. Therefore the second term of the sum in ( $1$ ) is even.

$C a s e 2 :$ $n$ is odd. Since $d_{u}$ is odd, we have ${\bar{d}}_{u} = n - 1 - d_{u}$ is odd. And in this case $d_{u}^{2} {\bar{d}}_{u}$ 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 $L z (G)$ is even.

The union of two graphs $G$ and $H$ denoted by $G \cup H$ is the graph with vertex set $V (G) \cup V (H)$ and edge set $E (G) \cup E (H)$ . 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 $K_{2} \cup K_{1}$ are the graphs with second minimum Lanzhou index $2$ .

Proposition 2. For any graph $G$ , $L z (G) = 2$ if and only if $G ≅ P_{3}$ or $G ≅ K_{2} \cup K_{1}$ .

Proof. For $G ≅ P_{3}$ or $G ≅ K_{2} \cup K_{1}$ , we have $L z (G) = 2$ . Moreover, $L z (K_{n}) = 0 = L z ({\bar{K}}_{n})$ . Otherwise, we have $n \geq 4$ and there exist two vertices $v$ and $w$ in $G$ such that $1 \leq d_{v} \leq n - 2$ and $1 \leq d_{w} \leq n - 2$ . Then $L z (G) = \sum_{u \in V (G)} d_{u}^{2} (n - 1 - d_{u}) \geq d_{v}^{2} (n - 1 - d_{v}) + d_{w}^{2} (n - 1 - d_{w}) > 2 as n \geq 4.$ 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 $0 \leq L z (G) \leq \frac{4}{27} n (n - 1)^{3} .$

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 $Δ$ .

Theorem 2. Let $G$ be a connected graph with $n$ vertices and $m$ edges having minimum degree $δ$ and maximum degree $Δ$ . Then $(n - 1) δ - Δ^{2} \leq \frac{L z (G)}{2 m} \leq (n - 1) Δ - δ^{2}$ with both equalities hold if and only if $G$ is a regular graph.

Proof. Since $δ \leq d_{u} \leq Δ$ , by definition of Lanzhou index, we have $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} (d_{u} {\bar{d}}_{u} + d_{v} {\bar{d}}_{v}) \\ = & \sum_{u v \in E (G)} ((n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2})) \\ \geq & \sum_{u v \in E (G)} ((n - 1) 2 δ - 2 Δ^{2}) \\ = & 2 m ((n - 1) δ - Δ^{2}) . \end{array}$ From the above result, we get the lower bound. Moreover, the equality holds in the lower bound if and only if $d_{u} = d_{v} = δ = Δ$ for any edge $u v \in E (G)$ , that is, if and only if $G$ is a regular graph. Again since $δ \leq d_{u} \leq Δ$ , similarly, we can get the upper bound on the Lanzhou index of graph $G$ . Moreover, the right equality holds if and only if $G$ is a regular graph.

Theorem 3. Let $G$ be a graph of order $n$ with $m$ edges. Then $L z (G) + L z (\bar{G}) = (n - 1) \sum_{u \in V (G)} d_{u} {\bar{d}}_{u} = (n - 1) [2 (n - 1) m - M_{1} (G)],$ where $M_{1} (G)$ is the first Zagreb index of graph $G$ .

Proof. We have $\begin{array}{rcl} L z (G) + L z (\bar{G}) & = & \sum_{u \in V (G)} d_{u}^{2} (n - 1 - d_{u}) + \sum_{u \in V (G)} {\bar{d}}_{u}^{2} (n - 1 - {\bar{d}}_{u}) \\ = & \sum_{u \in V (G)} d_{u}^{2} (n - 1 - d_{u}) + \sum_{u \in V (G)} (n - 1 - d_{u})^{2} d_{u} \\ = & \sum_{u \in V (G)} (n - 1 - d_{u}) d_{u} ((n - 1 - d_{u}) + d_{u}) \\ = & (n - 1) \sum_{u \in V (G)} [(n - 1) d_{u} - d_{u}^{2}] \\ = & (n - 1) [2 (n - 1) m - M_{1} (G)] . \end{array}$

Corollary 1. For self complementary graph $G$ , $L z (G) + L z (\bar{G}) = (n - 1) M_{1} (G)$ .

Theorem 4. Let $G$ be a graph with $n$ vertices and $e = u v$ be an edge in $G$ . If $H = G - e$ , then $L z (G) - L z (H) = (2 n + 1) (d_{u} + d_{v}) - 3 (d_{u}^{2} + d_{v}^{2}) - 2 n .$

Proof. By the definition of Lanzhou index, we have $\begin{array}{rcl} L z (G) & = & \sum_{x \in V (G) ∖ {u, v}} d_{x}^{2} (n - 1 - d_{x}) + d_{u}^{2} (n - 1 - d_{u}) + d_{v}^{2} (n - 1 - d_{v}), \\ L z (H) & = & \sum_{x \in V (G) ∖ {u, v}} d_{x}^{2} (n - 1 - d_{x}) + (d_{u} - 1)^{2} (n - d_{u}) + (d_{v} - 1)^{2} (n - d_{v}) . \end{array}$

Therefore we have $\begin{array}{rcl} L z (G) - L z (H) & = & - (d_{u}^{2} + d_{v}^{2}) + (2 d_{u} - 1) (n - d_{u}) + (2 d_{v} - 1) (n - d_{v}) \\ = & (2 n + 1) (d_{u} + d_{v}) - 3 (d_{u}^{2} + d_{v}^{2}) - 2 n . \end{array}$ This completes the proof of the theorem.

Let $T_{n}$ be a tree with $n$ vertices and $T_{n}^{Δ}$ denotes the set of all trees on $n$ vertices with maximum degree at most $Δ$ . Vukičević et al. [4] obtained the following result:

Proposition 4. [4] Let $n \geq 8$ be an integer and $T_{n} \in T_{n}^{4}$ . Then $4 n^{2} - 18 n + 20 \leq L z (T_{n}) \leq 6 n^{2} + O (n) .$

Theorem 5. Let $G$ be a unicyclic graph with $n (\geq 8)$ vertices and $Δ (G) = 4$ . Then $L z (G) \leq 6 n^{2} + O (n) .$

Proof. Let $e = u v$ be an edge in $G$ such that $H = G - e$ , where $H$ is a tree of order $n$ . Then by Theorem 1, we see that $L z (G) \leq L z (H) + (2 n + 1) (d_{u} + d_{v}),$ where $d_{u}$ and $d_{v}$ are the degrees of $u$ and $v$ , respectively.

Since $d_{u}, d_{v} \leq 4$ , from the above result, we have $L z (G) \leq L z (H) + 8 (2 n + 1) .$ Since $H$ is a tree, by Proposition 1, we obtain $L z (G) \leq 6 n^{2} + O (n) .$ This completes the proof of the theorem.

3. Relationships between Lanzhou index and other topological indices

The eccentricity of a vertex $v$ in a graph $G$ is defined as $e c c_{G} (v) = m a x {d_{G} (v, u) | u \in V (G)}$ . The radius of a graph $G$ , denoted by $r (G)$ is $r (G) = m i n {e c c_{G} (v) | v \in V (G)} .$

Lemma 1. [19] Let $G$ be a nontrivial connected graph of order $n$ . For each vertex $v$ in $G$ , it holds $e c c_{G} (v) \leq n - d_{v} .$ Moreover, the above equality holds together for all vertices in $G$ if and only if $G ≅ P_{4}$ or $G ≅ K_{n} - i K_{2}$ $(0 \leq i \leq ⌊ \frac{n}{2} ⌋)$ , where $K_{n} - i K_{2}$ denotes the graph obtained by removing $i$ independent edges from $K_{n}$ .

We now give a relation between $L z$ and $M_{1}$ .

Theorem 6. For any graph $G$ , $\begin{matrix} (2) & L z (G) \geq (r (G) - 1) M_{1} (G) . \end{matrix}$ Moreover, the equality holds in $(2)$ if and only if $G ≅ K_{n}$ or $G ≅ K_{n} - \frac{n}{2} K_{2}$ $(n$ is even $)$ .

Proof. For any vertex $v$ in $G$ we have $n - d_{v} \geq e c c_{G} (v) \geq r (G)$ . Now by the definition of Lanzhou index, we have $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} (d_{u} (n - 1 - d_{u}) + d_{v} (n - 1 - d_{v})) \\ \geq & \sum_{u v \in E (G)} (r (G) - 1) (d_{u} + d_{v}) \\ = & (r (G) - 1) \sum_{u v \in E (G)} (d_{u} + d_{v}) \\ = & (r (G) - 1) M_{1} (G) . \end{array}$

Suppose that equality holds in $(2)$ . Then we have $n - d_{v} = e c c_{G} (v) = r (G)$ for all $v \in V (G)$ . This implies that $G$ is self-centered graph. By Lemma 1, we have $G ≅ K_{n} - i K_{2}$ $(0 \leq i \leq ⌊ \frac{n}{2} ⌋)$ . Hence $G ≅ K_{n}$ or $G ≅ K_{n} - \frac{n}{2} K_{2}$ ( $n$ is even).

Conversely, one can easily see that the equality holds in $(2)$ for $K_{n}$ or for $K_{n} - \frac{n}{2} K_{2}$ ( $n$ is even).

Here we give a relation between $L z$ , $M_{1}$ and $M_{2}$ .

Theorem 7. Let $G$ be a graph of order $n$ . If $d_{u} + d_{v} \geq n$ for each edge $u v \in E (G)$ , then $L z (G) \leq 2 M_{2} (G) - M_{1} (G) .$

Proof. From the definition of Lanzhou index with the given condition, we have $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} (d_{u} (n - 1 - d_{u}) + d_{v} (n - 1 - d_{v})) \\ \leq & \sum_{u v \in E (G)} (d_{u} (d_{v} - 1) + d_{v} (d_{u} - 1)) \\ = & 2 \sum_{u v \in E (G)} d_{u} d_{v} - \sum_{u v \in E (G)} (d_{u} + d_{v}) \\ = & 2 M_{2} (G) - M_{1} (G) . \end{array}$

We now mention two more relations between $L z$ , $M_{1}$ and $M_{2}$ .

Theorem 8. Let $G$ be a graph of order $n$ . Then $L z (G) \leq (n - 1) M_{1} (G) - 2 M_{2} (G)$ with equality holding if and only if each connected component of $G$ is regular. Moreover, $(n - 1 - Δ) M_{1} (G) \leq L z (G) \leq (n - 1 - δ) M_{1} (G)$ with both equalities hold if and only if $G$ is a regular graph.

Proof. One can easily see that $F (G) \geq 2 M_{2} (G)$ with equality holding if and only if each connected component of $G$ is regular, and $δ M_{1} (G) \leq F (G) \leq Δ M_{1} (G)$ with both equalities hold if and only if $G$ is a regular graph. Since $L z (G) = (n - 1) M_{1} (G) - F (G)$ , 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 $ξ^{c} (G) = \sum_{u \in V (G)} d_{u} e c c_{G} (u) .$

Here we give a relation between $L z$ , $M_{1}$ and $ξ^{c}$ .

Theorem 9. Let $G$ be a graph of order $n$ and minimum degree $δ$ . Then $L z (G) \geq δ ξ^{c} (G) - M_{1} (G)$ with equality holding if and only if $G ≅ K_{n}$ or $G ≅ K_{n} - \frac{n}{2} K_{2}$ $(n$ is even $)$ .

Proof. From the definition of Lanzhou index, we have $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} (d_{u} (n - 1 - d_{u}) + d_{v} (n - 1 - d_{v})) \\ \geq & \sum_{u v \in E (G)} (d_{u} (e c c_{G} (u) - 1) + d_{v} (e c c_{G} (v) - 1)) \\ = & \sum_{u v \in E (G)} (d_{u} e c c_{G} (u) + d_{v} e c c_{G} (v)) - \sum_{u v \in E (G)} (d_{u} + d_{v}) \\ \geq & δ \sum_{u \in V (G)} d_{u} e c c_{G} (u) - M_{1} (G) = δ ξ^{c} (G) - M_{1} (G) . \end{array}$ By the proof of the Theorem 6, one can easily see that the equality holds if and only if $G ≅ K_{n}$ or $G ≅ K_{n} - \frac{n}{2} K_{2}$ $(n$ is even $)$ .

We now give a relation between $L z$ and $M_{2}$ .

Theorem 10. Let $G$ be a connected graph with $n$ vertices and minimum degree $δ$ . Then $\frac{2 (n - 2)}{Δ} M_{2} (G) \leq L z (G) \leq \frac{2 (n - 2)}{δ} M_{2} (G)$ 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 $\begin{array}{rcl} g (x, y) & = & \frac{(n - 1) (x + y) - (x^{2} + y^{2})}{x y} \\ = & (n - 1) (\frac{1}{y} + \frac{1}{x}) - (\frac{x}{y} + \frac{y}{x}), where δ \leq x \leq y \leq Δ \leq n - 1. \end{array}$

Now, $\begin{array}{rcl} \frac{\partial g (x, y)}{\partial x} & = & (n - 1) (- \frac{1}{x^{2}}) - (\frac{1}{y} - \frac{y}{x^{2}}) \\ = & - \frac{1}{x^{2}} (n - 1 - y) - \frac{1}{y} < 0, for δ \leq x \leq y \leq Δ \leq n - 1. \end{array}$

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 $(Δ, Δ)$ . Hence $g (δ, δ) \leq \frac{2 (n - 2)}{δ} and g (Δ, Δ) \geq \frac{2 (n - 2)}{Δ} .$

This implies that $(n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2}) \geq \frac{2 (n - 2)}{Δ} d_{u} d_{v}$ and $(n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2}) \leq \frac{2 (n - 2)}{δ} d_{u} d_{v} .$

Using the above results with the definition of the Lanzhou index, we have $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} ((n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2})) \\ \geq & \sum_{u v \in E (G)} \frac{2 (n - 2)}{Δ} d_{u} d_{v} = \frac{2 (n - 2)}{Δ} M_{2} (G) \end{array}$ and $L z (G) = \sum_{u v \in E (G)} ((n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2})) \leq \frac{2 (n - 2)}{δ} M_{2} (G)$

Moreover, both equalities hold in $(2)$ if and only if $d_{u} = d_{v} = δ$ or $d_{u} = d_{v} = Δ$ for any edge $u v \in E (G)$ . Since $G$ is connected, both equalities hold in $(2)$ if and only if $G$ is a regular graph.

The irregularity of a graph $G$ , denoted by $i r r (G)$ is defined as $i r r (G) = \sum_{u v \in E (G)} | d_{u} - d_{v} | .$ 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 $L z$ with $M_{1} (G), M_{2} (G), i r r (G)$ and $F (G)$ of graph $G$ .

Theorem 11. Let $G$ be a connected graph with $n$ vertices. Then $(n - 1) M_{1} (G) \leq i r r (G)^{2} + L z (G) + 2 M_{2} (G) \leq (n - 1) M_{1} (G) + 2 (\binom{m}{2}) (Δ - δ)^{2},$ with left $(r i g h t)$ 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 $\begin{array}{rcl} i r r (G)^{2} & = & (\sum_{u v \in E (G)} | d_{u} - d_{v} |)^{2} \\ = & \sum_{u v \in E (G)} (d_{u} - d_{v})^{2} + 2 \sum_{\binom{u v, x y \in E (G)}{u v \neq x y}} | d_{u} - d_{v} | | d_{x} - d_{y} | \\ \leq & \sum_{u v \in E (G)} (d_{u}^{2} + d_{v}^{2}) - 2 \sum_{u v \in E (G)} d_{u} d_{v} + 2 (\binom{m}{2}) (Δ - δ)^{2} \\ = & (n - 1) M_{1} (G) - L z (G) - 2 M_{2} (G) + 2 (\binom{m}{2}) (Δ - δ)^{2} \end{array}$ as $L z (G) = (n - 1) M_{1} (G) - F (G)$ . Hence we get the right inequality. Moreover, the equality holds if and only if $| d_{u} - d_{v} | = Δ - δ$ for all $u v \in E (G)$ , that is, if and only if $G$ is a regular graph or a bipartite semiregular graph as $G$ is connected.

Now, $\begin{array}{rcl} i r r (G)^{2} & = & (\sum_{u v \in E (G)} | d_{u} - d_{v} |)^{2} \\ \geq & \sum_{u v \in E (G)} (d_{u} - d_{v})^{2} \\ = & \sum_{u v \in E (G)} (d_{u}^{2} + d_{v}^{2}) - 2 \sum_{u v \in E (G)} d_{u} . d_{v} \\ = & F (G) - 2 M_{2} (G) \\ = & (n - 1) M_{1} (G) - L z (G) - 2 M_{2} (G) \end{array}$ as $L z (G) = (n - 1) M_{1} (G) - F (G)$ . Hence we get the left inequality. Moreover, the left equality holds if and only if $d_{u} = d_{v}$ for all edges $u v \in E (G)$ , 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 $S I (G) = \frac{1}{2} \sum_{{u, v} \subseteq V (G)} (d_{u} + d_{v}) d (u, v),$ 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 $G \lor H$ of two simple graphs $G$ and $H$ is the graph with the vertex set $V (G \lor H) = V (G) \cup V (H)$ and the edge set $E (G \lor H) = E (G) \cup E (H) \cup {u v : u \in V (G), v \in V (H)}$ . The following theorem gives a relation between Schultz index and Lanzhou index.

Theorem 12. Let $G$ be a connected graph of order $n$ . Then $S I (G) \geq \frac{1}{2 (n - 1)} (L z (G) + F (G)) + (n (n - 1) - 2 m) δ$ with equality holding if and only if $G ≅ K_{n}$ or $G ≅ K_{δ} \lor (n - δ) K_{1}$ $(δ < n - 1)$ or $G$ is a regular graph with diameter 2.

Proof. By definition of Schultz index we have $\begin{array}{rcl} S I (G) & = & \frac{1}{2} \sum_{{u, v} \subseteq V (G)} (d_{u} + d_{v}) d (u, v) \\ = & \frac{1}{2} \sum_{u v \in E (G)} (d_{u} + d_{v}) + \frac{1}{2} \sum_{{u, v} \subseteq V (G), d (u, v) \geq 2} (d_{u} + d_{v}) d (u, v) \\ \geq & \frac{1}{2} M_{1} (G) + \sum_{{u, v} \subseteq V (G), d (u, v) \geq 2} (d_{u} + d_{v}) \\ \geq & \frac{1}{2} M_{1} (G) + 2 ((\binom{n}{2}) - m) δ \\ = & \frac{1}{2 (n - 1)} (L z (G) + F (G)) + (n (n - 1) - 2 m) δ . \end{array}$ 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 $δ$ . If $Δ = n - 1$ , then $G ≅ K_{n}$ or $G ≅ K_{δ} \lor (n - δ) K_{1}$ $(δ < n - 1)$ . Otherwise, $Δ < n - 1$ . Then all the vertices in $G$ are of degree $δ$ . Hence $G$ is a regular graph with diameter 2.

Conversely, let $G ≅ K_{n}$ . Then $L z (G) = 0$ , $F (G) = n (n - 1)^{3}$ and $S I (G) = n (n - 1)^{2} / 2$ . Hence the equality holds.

Let $G ≅ K_{δ} \lor (n - δ) K_{1}$ $(δ < n - 1)$ . Then $d_{1} = d_{2} = \dots = d_{δ} = n - 1$ and $d_{δ + 1} = d_{δ + 2} = \dots = d_{n} = δ$ . Then $L z (G) + F (G) = (n - 1)^{3} δ + (n - 1) (n - δ) δ^{2}, 2 m = δ (n - 1) + (n - δ) δ$ and $S I (G) = \frac{δ (δ - 1) (n - 1)}{2} + (n - δ) (n - δ - 1) δ + \frac{δ (n - δ) (n - 1 + δ)}{2} .$ Now, $\begin{array}{rcl} \frac{1}{2 (n - 1)} (L z (G) + F (G)) + (n (n - 1) - 2 m) δ \\ = & \frac{δ (n - 1)^{2} + (n - δ) δ^{2}}{2} + (n (n - 1) - δ (n - 1) - (n - δ) δ) δ \\ = & \frac{δ (δ - 1) (n - 1)}{2} + (n - δ) (n - δ - 1) δ + \frac{δ (n - δ) (n - 1 + δ)}{2} \\ = & S I (G) . \end{array}$

Let $G$ be a regular graph with diameter 2. Then $d_{1} = d_{2} = \dots = d_{n} = δ$ . Then $L z (G) + F (G) = (n - 1) n δ^{2}, 2 m = n δ and S I (G) = δ [m + (\frac{n (n - 1)}{2} - m) 2] .$ Now, $\begin{array}{rcl} \frac{1}{2 (n - 1)} (L z (G) + F (G)) + (n (n - 1) - 2 m) δ \\ = & \frac{n δ^{2}}{2} + (n (n - 1) - n δ) δ \\ = & S I (G) . \end{array}$ 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 $I S I (G) = \sum_{u v \in E (G)} \frac{1}{\frac{1}{d_{u}} + \frac{1}{d_{v}}} .$

Theorem 13. Let $G$ be a graph with $n$ vertices and maximum degree $Δ$ . Then $L z (G) \geq 4 (n - 1 - Δ) I S I (G)$ with equality holding if and only if $G$ is a regular graph.

Proof. Since $Δ$ is the maximum degree in $G$ , we have $d_{u} (Δ - d_{u}) + d_{v} (Δ - d_{v}) \geq 0, that is, \frac{d_{u}^{2} + d_{v}^{2}}{d_{u} + d_{v}} \leq Δ$ with equality holding if and only if $d_{u} = d_{v} = Δ$ .

One can easily see that $(d_{u} - d_{v})^{2} \geq 0, that is, (d_{u} + d_{v})^{2} \geq 4 d_{u} d_{v}, that is, (d_{u} + d_{v}) \geq \frac{4 d_{u} d_{v}}{d_{u} + d_{v}} = \frac{4}{\frac{1}{d_{u}} + \frac{1}{d_{v}}}$ with equality holding if and only if $d_{u} = d_{v}$ .

Using the above results with the definition of the Lanzhou index, we have $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} ((n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2})) \\ = & \sum_{u v \in E (G)} (d_{u} + d_{v}) (n - 1 - \frac{d_{u}^{2} + d_{v}^{2}}{d_{u} + d_{v}}) \\ \geq & \sum_{u v \in E (G)} (d_{u} + d_{v}) (n - 1 - Δ) \\ \geq & 4 (n - 1 - Δ) \sum_{u v \in E (G)} \frac{1}{\frac{1}{d_{u}} + \frac{1}{d_{v}}} \\ = & 4 (n - 1 - Δ) I S I (G) . \end{array}$ Moreover, the equality holds if and only if $d_{u} = d_{v} = Δ$ for any edge $u v \in E (G)$ , that is, if and only if $G$ is a regular graph.

Corollary 1. Let $G$ be a graph with $n$ vertices, $m$ edges and maximum degree $Δ$ . Then $L z (G) \geq \frac{4 m^{2} (n - 1 - Δ)}{n}$ with equality holding if and only if $G$ is a regular graph.

Proof. Since $I S I (G) \geq \frac{m^{2}}{n}$ , 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, $S D D$ , was defined in [17] as $S D D (G) = \sum_{u v \in E (G)} \frac{d_{u}^{2} + d_{v}^{2}}{d_{u} d_{v}} .$ For recent results on $S D D (G)$ see the papers [25,26,27,28,29] and the references cited therein. Here we give a relation between $L z$ and $S D D$ .

Theorem 14. Let $G$ be a graph with $n$ vertices and maximum degree $Δ$ . Then $L z (G) \leq Δ^{2} (n (n - 1) - S D D (G))$ with equality holding if and only if $G$ is a regular graph.

Proof. Let $G ≅ H \cup p K_{1}$ $(p \geq 0)$ , where $H$ is a graph of order $n - p$ . Since $n - 1 - d_{u} \geq 0$ for all $u \in V (G)$ , by the definition of Lanzhou index, we obtain $\begin{array}{rcl} L z (G) & = & \sum_{u v \in E (G)} [(n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2})] \\ = & \sum_{u v \in E (G)} d_{u} d_{v} \frac{(n - 1) (d_{u} + d_{v}) - (d_{u}^{2} + d_{v}^{2})}{d_{u} d_{v}} \\ \leq & Δ^{2} \sum_{u v \in E (G)} (\frac{(n - 1) (d_{u} + d_{v})}{d_{u} d_{v}} - \frac{d_{u}^{2} + d_{v}^{2}}{d_{u} d_{v}}) \\ = & Δ^{2} [(n - 1) \sum_{u v \in E (G)} (\frac{1}{d_{u}} + \frac{1}{d_{v}}) - \sum_{u v \in E (G)} \frac{d_{u}^{2} + d_{v}^{2}}{d_{u} d_{v}}] \\ = & Δ^{2} ((n - p) (n - 1) - S D D (G)) \\ \leq & Δ^{2} (n (n - 1) - S D D (G)) . \end{array}$ The first part of the proof is done.

The equality holds if and only if $d_{u} = d_{v} = Δ$ for any edge $u v \in E (H)$ and $p = 0$ , that is, if and only if $G$ is a regular graph.

4. The join and corona of graphs

The following theorem provides the formula to find the Lanzhou index of the join of two graphs.

Theorem 15. Let $G$ and $H$ be two graphs. Then $\begin{array}{rcl} L z (G \lor H) = L z (G) + L z (H) + 4 n_{2} (n_{1} - 1) | E (G) | + 4 n_{1} (n_{2} - 1) | E (H) | - 2 n_{2} M_{1} (G) \\ - 2 n_{1} M_{1} (H) + n_{1} n_{2} (2 n_{1} n_{2} - n) - 2 n_{2}^{2} | E (G) | - 2 n_{1}^{2} | E (H) | . \end{array}$

Proof. Let $G$ and $H$ be two graphs with $n_{1}$ and $n_{2}$ vertices. Then $G \lor H$ has $n_{1} + n_{2}$ vertices. We have $d_{G \lor H} (u) = d_{G} (u) + n_{2}$ , for $u \in V (G)$ and $d_{G \lor H} (u) = d_{H} (u) + n_{1}$ , for $u \in V (H)$ . By the definition of Lanzhou index, we obtain $\begin{array}{rcl} L z (G \lor H) & = & \sum_{u \in V (G \lor H)} d_{u}^{2} (n_{1} + n_{2} - 1 - d_{u}) \\ = & \sum_{u \in V (G)} d_{G \lor H} (u)^{2} (n_{1} + n_{2} - 1 - d_{G \lor H} (u)) \\ + \sum_{u \in V (H)} d_{G \lor H} (u)^{2} (n_{1} + n_{2} - 1 - d_{G \lor H} (u)) . \end{array}$ Now, $\begin{array}{rcl} \sum_{u \in V (G)} d_{G \lor H} (u)^{2} (n_{1} + n_{2} - 1 - d_{G \lor H} (u)) \\ = & \sum_{u \in V (G)} (d_{G} (u) + n_{2})^{2} (n_{1} - 1 - d_{G} (u)) \\ = & \sum_{u \in V (G)} d_{G} (u)^{2} {\bar{d}}_{G} (u) + 2 n_{2} \sum_{u \in V (G)} d_{G} (u) (n_{1} - 1 - d_{G} (u)) + n_{2}^{2} \sum_{u \in V (G)} (n_{1} - 1 - d_{G} (u)) \\ = & L z (G) + 4 n_{2} (n_{1} - 1) | E (G) | - 2 n_{2} M_{1} (G) + n_{2}^{2} (n_{1} (n_{1} - 1) - 2 | E (G) |) . \end{array}$ Similarly, we obtain $\begin{array}{rcl} \sum_{u \in V (H)} d_{G \lor H} (u)^{2} (n_{1} + n_{2} - 1 - d_{G \lor H} (u)) \\ = & L z (H) + 4 n_{1} (n_{2} - 1) | E (H) | - 2 n_{1} M_{1} (H) + n_{1}^{2} (n_{2} (n_{2} - 1) - 2 | E (H) |) \end{array}$ Combining two above results, we get the required result.

The corona product $G \circ H$ of two graphs G and H is defined to be the graph $Γ$ obtained by taking one copy of $G$ (which has $n_{1}$ vertices) and $n_{1}$ copies of $H$ (which has $n_{2}$ vertices), and then joining the $i$ -th vertex of $G$ to every vertex in the $i$ -th copy of $H$ , $i = 1, 2, \dots, n_{1}$ . The next theorem provides the Lanzhou index of the corona product of two graphs $G$ and $H$ .

Theorem 16. Let $G$ and $H$ be two graphs of order $n_{1}$ and $n_{2}$ , respectively. Then
$L z (G \circ H) = L z (G) + n_{1} L z (H) + A + n_{1} B,$
where
$A = 2 n_{2} (2 n_{1} n_{2} + 2 n_{1} - 3 n_{2} - 2) | E (G) | - 3 n_{2} M_{1} (G) + n_{1} n_{2}^{2} (n_{1} - 1) (n_{2} + 1)$
and
$B = 2 (2 n_{1} n_{2} + 2 n_{1} - 5) | E (H) | - 3 M_{1} (H) + n_{2} (n_{1} n_{2} + n_{1} - 2) .$

Proof. For any vertex $u \in V (G)$ , we have $d_{G \circ H} (u) = d_{G} (u) + n_{2}$ and for any vertex $u \in V (H)$ , we have $d_{G \circ H} (u) = d_{H} (u) + 1$ . Now, $\begin{array}{rcl} \sum_{u \in V (G)} (d_{G \circ H} (u))^{2} (n_{1} n_{2} + n_{1} - 1 - d_{G \circ H} (u)) \\ = & \sum_{u \in V (G)} (d_{G} (u) + n_{2})^{2} (n_{1} n_{2} + n_{1} - 1 - d_{G} (u) - n_{2}) \\ = & \sum_{u \in V (G)} [d_{G} (u)^{2} {\bar{d}}_{G} (u) - 3 n_{2} d_{G} (u)^{2} + n_{2} (2 n_{1} n_{2} + 2 n_{1} - 3 n_{2} - 2) d_{G} (u) \\ + n_{2}^{2} (n_{1} - 1) (n_{2} + 1)] \\ = & L z (G) - 3 n_{2} M_{1} (G) + 2 n_{2} (2 n_{1} n_{2} + 2 n_{1} - 3 n_{2} - 2) | E (G) | + n_{1} n_{2}^{2} (n_{1} - 1) (n_{2} + 1) \end{array}$ and $\begin{array}{rcl} \sum_{u \in V (H)} (d_{G \circ H} (u))^{2} (n_{1} n_{2} + n_{1} - 1 - d_{G \circ H} (u)) \\ = & \sum_{u \in V (H)} (d_{H} (u) + 1)^{2} (n_{1} n_{2} + n_{1} - 2 - d_{H} (u)) \\ = & \sum_{u \in V (H)} [d_{H} (u)^{2} {\bar{d}}_{H} (u) - 3 d_{H} (u)^{2} + (2 n_{1} n_{2} + 2 n_{1} - 5) d_{H} (u) + n_{1} n_{2} + n_{1} - 2] \\ = & L z (H) - 3 M_{1} (H) + 2 (2 n_{1} n_{2} + 2 n_{1} - 5) | E (H) | + n_{2} (n_{1} n_{2} + n_{1} - 2) . \end{array}$ Using the above results, we obtain $\begin{array}{rcl} L z (G \circ H) & = & \sum_{u \in V (G \circ H)} (d_{G \circ H} (u))^{2} (n_{1} n_{2} + n_{1} - 1 - d_{G \circ H} (u)) \\ = & \sum_{u \in V (G)} (d_{G} (u) + n_{2})^{2} (n_{1} n_{2} + n_{1} - 1 - d_{G} (u) - n_{2}) \\ + \sum_{u \in V (G)} \sum_{v \in V (H)} (d_{H} (v) + 1)^{2} (n_{1} n_{2} + n_{1} - 2 - d_{H} (v)) \\ = & L z (G) + n_{1} L z (H) + A + n_{1} B, \end{array}$ where
$A = 2 n_{2} (2 n_{1} n_{2} + 2 n_{1} - 3 n_{2} - 2) | E (G) | - 3 n_{2} M_{1} (G) + n_{1} n_{2}^{2} (n_{1} - 1) (n_{2} + 1)$
and
$B = 2 (2 n_{1} n_{2} + 2 n_{1} - 5) | E (H) | - 3 M_{1} (H) + n_{2} (n_{1} n_{2} + n_{1} - 2) .$ This completes the proof of the theorem.

Acknowledgements

The authors would like to thank the referee for his/her valuable comments which led to an improvement of the original manuscript. The first author gratefully acknowledges support by National Natural Science Foundation of China (61320106005 and 61772214) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (154200510012). The second author was supported by the Sungkyun research fund, Sungkyunkwan University, 2017, and National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642.

Conflict of Interest

The authors declare no conflict of interests.

References:

Gutman, I. and Trinajstic, N., 1972. Graph theory and molecular orbitals. Total $π$ -electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4), pp.535-538.[Google Scholor]
Todeschini, R. and Consonni, V., 2008. Handbook of molecular descriptors. John Wiley & Sons.[Google Scholor]
Furtula, B. and Gutman, I., 2015. A forgotten topological index. Journal of Mathematical Chemistry, 53(4), pp.1184-1190.[Google Scholor]
Vukičević, D., Li, Q., Sedlar, J. and Došlić, T., 2018. Lanzhou index. MATCH Communications in Mathematical and in Computer Chemistry, 80(3), pp.863-876.[Google Scholor]
Došlić, T., Furtula, B., Graovac, A., Gutman, I., Moradi, S. and Yarahmadi, Z., 2011. On vertex-degree-based molecular structure descriptors. MATCH Communications in Mathematical and in Computer Chemistry, 66, 616–626.[Google Scholor]
Došlić, T., Réti, T., & Vukičević, D., 2011. On the vertex degree indices of connected graphs. Chemical Physics Letters, 512(4-6), pp.283-286.[Google Scholor]
Das, K.C. and Gutman, I., 2004. Some properties of the second Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry, 52(1), pp.103–112.[Google Scholor]
Das, K.C., Gutman, I. and Zhou, B., 2009. New upper bounds on Zagreb indices. Journal of Mathematical Chemistry, 46(2), pp.514-521.[Google Scholor]
Furtula, B., Gutman, I. and Dehmer, M., 2013. On structure-sensitivity of degree-based topological indices. Applied Mathematics and Computation, 219(17), pp.8973-8978.[Google Scholor]
Gutman, I., 2013. Degree-based topological indices. Croatica Chemica Acta, 86(4), pp.351-361.[Google Scholor]
Gutman, I. and Došlić, J., 2013. Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices. Journal of the Serbian Chemical Society, 78(6), pp.805-810.[Google Scholor]
Hua, H., 2008. Zagreb $M_{1}$ index, independence number and connectivity in graphs.MATCH Communications in Mathematical and in Computer Chemistry, 60, pp.45-56.[Google Scholor]
Hua, H. and Das, K.C., 2013. The relationship between the eccentric connectivity index and Zagreb indices. Discrete Applied Mathematics, 161(16-17), pp.2480-2491.[Google Scholor]
Milovanović, E.I., Milovanović, I. Ž. and Matejić, M.M., 2018. Remark on spectral study of the geometric-arithmetic index and some generalizations. Applied Mathematics and Computation, 334, pp.206-213.[Google Scholor]
Milovanovic, I.Ž., Milovanović, E.I. and Matejić, M.M., 2018. On upper bounds for the geometric-arithmetic topological index. MATCH Communications in Mathematical and in Computer Chemistry, 80, pp.109-127.[Google Scholor]
Vukićević, D. and Gašperov, M., 2010. Bond additive modeling 1. Adriatic indices. Croatica Chemica Acta, 83(3), p.2[Google Scholor]
Vukicevic, D., 2010. Bond additive modeling 2. Mathematical properties of max-min rodeg index. Croatica Chemica Acta, 83(3), pp.261-273.[Google Scholor]
Sharma, V., Goswami, R. and Madan, A.K., 1997. Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies. Journal of chemical information and computer sciences, 37(2), pp.273-282.[Google Scholor]
Ilić, A., Yu, G. and Feng, L., 2011. On the eccentric distance sum of graphs. Journal of Mathematical Analysis and Applications, 381(2), pp.590-600.[Google Scholor]
Luo, W. and Zhou, B., 2010. On the irregularity of trees and unicyclic graphs with given matching number. Utilitas Mathematica, 83, 141–147.[Google Scholor]
Albertson, M.O., 1997. The irregularity of a graph. Ars Combinatoria, 46, pp.219-225.[Google Scholor]
Zhou, B. and Luo, W., 2008. On irregularity of graphs. Ars Combinatoria, 88, pp.55-64.[Google Scholor]
Schultz, H.P., 1989. Topological organic chemistry. 1. Graph theory and topological indices of alkanes. Journal of Chemical Information and Computer Sciences, 29(3), pp.227-228.[Google Scholor]
Karelson, M., 2000. Molecular descriptors in QSAR/QSPR. Wiley Interscience, New York (2000).[Google Scholor]
Das, K.C., Gutman, I. and Furtula, B., 2017. On spectral radius and energy of extended adjacency matrix of graphs. Applied Mathematics and Computation, 296, pp.116-123.[Google Scholor]
Das, K.C., Matejić, M., Milovanović, E. and Milovanović, I., 2019. Bounds for symmetric division deg index of graphs. Filomat, 33(3), pp.683-698.[Google Scholor]
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Vasilyev, A., 2014. Upper and lower bounds of symmetric division deg index. Iranian Journal of Mathematical Chemistry,5(2), 91–98.[Google Scholor]

[1] Gutman, I. and Trinajstic, N., 1972. Graph theory and molecular orbitals. Total $π$ -electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4), pp.535-538.[Google Scholor]

[2] Todeschini, R. and Consonni, V., 2008. Handbook of molecular descriptors. John Wiley & Sons.[Google Scholor]

[3] Furtula, B. and Gutman, I., 2015. A forgotten topological index. Journal of Mathematical Chemistry, 53(4), pp.1184-1190.[Google Scholor]

[4] Vukičević, D., Li, Q., Sedlar, J. and Došlić, T., 2018. Lanzhou index. MATCH Communications in Mathematical and in Computer Chemistry, 80(3), pp.863-876.[Google Scholor]

[5] Došlić, T., Furtula, B., Graovac, A., Gutman, I., Moradi, S. and Yarahmadi, Z., 2011. On vertex-degree-based molecular structure descriptors. MATCH Communications in Mathematical and in Computer Chemistry, 66, 616–626.[Google Scholor]

[6] Došlić, T., Réti, T., & Vukičević, D., 2011. On the vertex degree indices of connected graphs. Chemical Physics Letters, 512(4-6), pp.283-286.[Google Scholor]

[7] Das, K.C. and Gutman, I., 2004. Some properties of the second Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry, 52(1), pp.103–112.[Google Scholor]

[8] Das, K.C., Gutman, I. and Zhou, B., 2009. New upper bounds on Zagreb indices. Journal of Mathematical Chemistry, 46(2), pp.514-521.[Google Scholor]

[9] Furtula, B., Gutman, I. and Dehmer, M., 2013. On structure-sensitivity of degree-based topological indices. Applied Mathematics and Computation, 219(17), pp.8973-8978.[Google Scholor]

[10] Gutman, I., 2013. Degree-based topological indices. Croatica Chemica Acta, 86(4), pp.351-361.[Google Scholor]

[11] Gutman, I. and Došlić, J., 2013. Testing the quality of molecular structure descriptors. Vertex-degree-based topological indices. Journal of the Serbian Chemical Society, 78(6), pp.805-810.[Google Scholor]

[12] Hua, H., 2008. Zagreb $M_{1}$ index, independence number and connectivity in graphs.MATCH Communications in Mathematical and in Computer Chemistry, 60, pp.45-56.[Google Scholor]

[13] Hua, H. and Das, K.C., 2013. The relationship between the eccentric connectivity index and Zagreb indices. Discrete Applied Mathematics, 161(16-17), pp.2480-2491.[Google Scholor]

[14] Milovanović, E.I., Milovanović, I. Ž. and Matejić, M.M., 2018. Remark on spectral study of the geometric-arithmetic index and some generalizations. Applied Mathematics and Computation, 334, pp.206-213.[Google Scholor]

[15] Milovanovic, I.Ž., Milovanović, E.I. and Matejić, M.M., 2018. On upper bounds for the geometric-arithmetic topological index. MATCH Communications in Mathematical and in Computer Chemistry, 80, pp.109-127.[Google Scholor]

[16] Vukićević, D. and Gašperov, M., 2010. Bond additive modeling 1. Adriatic indices. Croatica Chemica Acta, 83(3), p.2[Google Scholor]

[17] Vukicevic, D., 2010. Bond additive modeling 2. Mathematical properties of max-min rodeg index. Croatica Chemica Acta, 83(3), pp.261-273.[Google Scholor]

[18] Sharma, V., Goswami, R. and Madan, A.K., 1997. Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies. Journal of chemical information and computer sciences, 37(2), pp.273-282.[Google Scholor]

[19] Ilić, A., Yu, G. and Feng, L., 2011. On the eccentric distance sum of graphs. Journal of Mathematical Analysis and Applications, 381(2), pp.590-600.[Google Scholor]

[20] Luo, W. and Zhou, B., 2010. On the irregularity of trees and unicyclic graphs with given matching number. Utilitas Mathematica, 83, 141–147.[Google Scholor]

[21] Albertson, M.O., 1997. The irregularity of a graph. Ars Combinatoria, 46, pp.219-225.[Google Scholor]

[22] Zhou, B. and Luo, W., 2008. On irregularity of graphs. Ars Combinatoria, 88, pp.55-64.[Google Scholor]

[23] Schultz, H.P., 1989. Topological organic chemistry. 1. Graph theory and topological indices of alkanes. Journal of Chemical Information and Computer Sciences, 29(3), pp.227-228.[Google Scholor]

[24] Karelson, M., 2000. Molecular descriptors in QSAR/QSPR. Wiley Interscience, New York (2000).[Google Scholor]

[25] Das, K.C., Gutman, I. and Furtula, B., 2017. On spectral radius and energy of extended adjacency matrix of graphs. Applied Mathematics and Computation, 296, pp.116-123.[Google Scholor]

[26] Das, K.C., Matejić, M., Milovanović, E. and Milovanović, I., 2019. Bounds for symmetric division deg index of graphs. Filomat, 33(3), pp.683-698.[Google Scholor]

[27] Furtula, B., Ch. Das, K. and Gutman, I., 2018. Comparative analysis of symmetric division deg index as potentially useful molecular descriptor. International Journal of Quantum Chemistry, 118(17), p.e25659.[Google Scholor]

[28] Gupta, C.K., Lokesha, V., Shwetha, S.B. and Ranjini, P.S., 2016. On the Symmetric Division deg Index of Graph. Southeast Asian Bulletin of Mathematics, 40(1), 59–80.[Google Scholor]

[29] Vasilyev, A., 2014. Upper and lower bounds of symmetric division deg index. Iranian Journal of Mathematical Chemistry,5(2), 91–98.[Google Scholor]

Contents

Ars Combinatoria

On Lanzhou Index of Graphs

Abstract

1. Introduction and Motivation

2. Some properties of Lanzhou index

3. Relationships between Lanzhou index and other topological indices

4. The join and corona of graphs

Acknowledgements

Conflict of Interest

References:

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