The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than m.


Introduction
All graphs considered in this paper are simple, undirected, and connected.Let G be a graph with vertex set V(G) = {v 1 , • • • , v n } and edge set E(G) = {e 1 , • • • , e m }.The degree of a vertex u ∈ V(G) is the number of edges incident to u, denoted by d G (u).The distance between two vertices u and v is the length of the shortest path connecting u and v, denoted by d G (u, v).The complement of G, denoted by G, is the graph with vertex set V(G) = V(G) and edge set E(G) = {uv : uv / ∈ E(G)}.A subgraph of G induced by H, denoted by H , is the subgraph of G that has the vertex set H, and for any two vertices u, v ∈ V(H), they are adjacent in H iff they are adjacent in G.The adjacency matrix of G is a square n × n matrix such that its element a ij is one when there is an edge from vertex u i to vertex u j , and zero when there is no edge, denoted by A(G).Let D(G) = diag(d 1 , d 2 , • • • , d n ) be the diagonal matrix of vertex degrees of G.The Laplacian matrix of G is defined as L(G) = D(G) − A(G), and the eigenvalues of L(G) are called Laplacian eigenvalues of G, denoted by It is well known that µ n = 0, and the multiplicity of zero corresponds to the number of connected components of G.
A bipartite graph is a graph whose vertex set can be partitioned into two disjoint sets U 1 and U 2 , such that each edge has an end vertex in U 1 and the other one in U 2 .A complete bipartite graph, denoted by K s,t , is a bipartite graph with |U 1 | = s and |U 2 | = t, where any two vertices u ∈ U 1 and v ∈ U 2 are adjacent.If every pair of distinct vertices in G is connected by a unique edge, we call G a complete graph.The complete graph with n vertices is denoted by K n .An independent set of G is a set of vertices, no two of which are adjacent.A graph G is called k-partite if its vertex-set can be partitioned into k different independent sets U 1 , • • • , U k .When k = 2, they are the bipartite graphs, and k = 3 the tripartite graphs.The vertex k-partiteness of graph G, denoted by v k (G), is the fewest number of vertices whose deletion from G yields a k-partite graph.A complete k-partite graph, denoted by where there is an edge between every pair of vertices from different independent sets.
A topological index is a numerical value that can be used to characterize some properties of molecule graphs in chemical graph theory.Recently, many researchers have paid much attention to studying different topological indices.Dimitrov [1] studied the structural properties of trees with minimal atom-bond connectivity index.Li and Fan [2] obtained the extremal graphs of the Harary index.Xu et al. [3] determined the eccentricity-based topological indices of graphs.Hayat et al. [4] studied the valency-based topological descriptors of chemical networks and their applications.Let G + uv be the graph obtained from G by adding an edge uv ∈ E(G).Let I(G) be a graph invariant, if I(G + uv) > I(G) (or I(G + uv) < I(G), respectively) for any edge uv ∈ E(G), then we call I(G) a monotonic increasing (or decreasing, respectively) graph invariant with the addition of edges [5].Let G n,m,k be the set of graphs with order n and vertex k-partiteness v k (G) ≤ m, where 1 ≤ m ≤ n − k.In [5][6][7], the authors have researched several monotonic topological indices in G n,m,2 , such as the Kirchhoff index, the spectral radius, the signless Laplacian spectral radius, the modified-Wiener index, the connective eccentricity index, and so on.Inspired by these results, we extend the parameter of graph partition from two-partiteness to arbitrary k-partiteness.Moreover, we study some monotonic topological indices and characterize the graphs with extremal monotonic topological indices in G n,m,k .

Preliminaries
The join of two-vertex-disjoint graphs The join operation can be generalized as follows.Let F = {G 1 , • • • , G k } be a set of vertex-disjoint graphs and H be an arbitrary graph with vertex set The H-join of the graphs V(G j ) and: If H = K 2 , the H-join is the usual join operation of graphs, and the complete k-partite graph where O s i is an empty graph of order s i , 1 ≤ i ≤ k.
For U ⊆ V(G), let G − U be the graph obtained from G by deleting the vertices in U and the edges incident with them.Lemma 1.Let G be an arbitrary graph in G n,m,k and I(G) be a monotonic increasing graph invariant.
Then, there exists k positive integers s Proof.Choose G ∈ G n,m,k with the maximum value of a monotonic increasing graph invariant such that Otherwise, there exists at least two vertices u ∈ U s i and v ∈ U s j for some i = j, which are not adjacent in G.By joining the vertices u and v, we get a new graph G + uv, obviously, G + uv ∈ G n,m,k .Then, I( G) < I( G + uv), which is a contradiction.
Secondly, we claim that U is the complete graph K m .Otherwise, there exists at least two vertices u, v ∈ U, which are not adjacent.By connecting the vertices u and v, we arrive at a new graph G + uv, obviously, G + uv ∈ G n,m,k .Then, we have I( G) < I( G + uv), a contradiction again.

Using a similar method, we can get
Without loss of generality, we assume that s 1 ≥ 2. By moving a vertex u ∈ O s 1 to the set of U and adding edges between u and all the other vertices in O s 1 , we get a new graph By the definition of the monotonic increasing graph invariant, we get I( G) < I( G), which is obviously another contradiction.
Therefore, G is the join of a complete graph with order m and a complete k-partite graph with order n − m.That is to say The proof of the lemma is completed.
Lemma 2. Let G be an arbitrary graph in G n,m,k and I(G) be a monotonic decreasing graph invariant.
Then, there exists k positive integers s

Main Results
In this section, we will characterize the graphs with an extremal monotonic increasing (or decreasing, respectively) graph invariant in G n,m,k .Assume that n − m = sk + t, where s is a positive integer and t is a non-negative integer with 0 ≤ t < k.

The Reformulated First Zagreb Index, Multiplicative-Sum Zagreb Index, and k-Partiteness
The first Zagreb index is used to analyze the structure-dependency of total π-electron energy on the molecular orbitals, introduced by Gutman and Trinajstć [8].It is denoted by: which can be also calculated as: Todeschini and Consonni [9] considered the multiplicative version of the first Zagreb index in 2010, defined as: For an edge e = uv ∈ E(G), we define the degree of e as Millicević et al. [10] introduced the reformulated first Zagreb index, defined as: Eliasi et al. [11] introduced another multiplicative version of the first Zagreb index, which is called the multiplicative-sum Zagreb index and defined as: They are widely used in chemistry to study the heat information of heptanes and octanes.For some recent results on the fourth Zagreb indices, one can see [12][13][14][15][16][17].
Theorem 1.Let G be a graph of order n > 2, and the join of a complete graph with order m and a complete k-partite graph with order n − m in Proof.By the definition of the reformulated first Zagreb index M 1 (G), we can calculate as follows: Therefore, , where s is a positive integer and t is a non-negative integer with 0 ≤ t < k.For simplicity, we write ).Then, the extremal value and the corresponding graph of the reformulated first Zagreb index M 1 (G) can be shown as follows.
Theorem 2. Let G be an arbitrary graph in G n,m,k .Then: with the equality holding if and only Proof.By Lemmas 1, 3, and Theorem 1, the extremal graph having the maximum reformulated first Zagreb index in G n,m,k is the graph Then, we obtain that: Theorem 3. Let G be a graph of order n > 2, and the join of a complete graph with order m and a complete k-partite graph with order n − m in Proof.By the definition of the multiplicative-sum Zagreb index Π * 1 (G), it is easy to see that: Hence, . By a simple calculation, we have: Theorem 4. Let G be an arbitrary graph in G n,m,k .Then: with the equality holding if and only if G ∼ = K m ∨ (K k [{k − t}O s , {s}O s+1 ]).

The General Laplacian-Energy-Like Invariant and k-Partiteness
The general Laplacian-energy-like invariant (also called the sum of powers of the Laplacian eigenvalues) of a graph G is defined by Zhou [18] as: where α is an arbitrary real number.S α (G) is the Laplacian-energy-like invariant [19], and the Laplacian energy [20] when α = 1 2 and α = 2, respectively.For α = −1, nS −1 (G) is equal to the Kirchhoff index [21], and α = 1, 1 2 S 1 (G) is equal to the number of edges in G.For some recent results on the general Laplacian-energy-like invariant, one can see [22][23][24][25].Lemma 5. [18] Let G be a graph with u, v ∈ It is well known that Laplacian eigenvalues of the complete graph K p are 0, p, • • • , p, and Laplacian eigenvalues of O p are 0, 0, • • • , 0. Then, the Laplacian eigenvalues of , where the multiplicity of s 2 + s 3 is s 1 − 1 and the multiplicity of By induction, we have that the Laplacian eigenvalues of From Lemma 6 and the above analysis, we obtain the following lemma.

Lemma 7.
Let G be a graph of order n, and the join of a complete graph with order m and a complete k-partite graph with order n , where the multiplicity of n is m + k − 1 and the multiplicity of n − s j is s j − 1, for 1 ≤ j ≤ k.
Theorem 5. Let G be a graph of order n > 2, and the join of a complete graph with order m and a complete k-partite graph with order n − m in Proof.By the definition of the general Laplacian-energy-like invariant S α (G) and Lemma 7, we conclude that: For α < 0, we have: where For 0 < α < 1, we have: where , and S α ( G) > S α ( G).
Theorem 6.Let G be an arbitrary graph in G n,m,k .Then, Proof.By Lemmas 1, 2, and Theorem 5, the extremal graph having the extremal value of the general Laplacian-energy-like invariant in G n,m,k should be the graph
Theorem 7. Let G be a graph of order n > 2, and the join of a complete graph with order m and a complete k-partite graph with order n − m in G n,m,k , i.e., G = Proof.By the definition of the general zeroth-order Randić index 0 R α (G), we have: For α < 0, we have: For 0 < α < 1, we have: where Proof.By Lemma 8 and Theorem 7, in view of Lemmas 1 and 2, the extremal graph having the extremal value of the general zeroth-order Randić index in G n,m,k should be the graph

The Modified-Wiener Index and k-Partiteness
The modified-Wiener index is defined by Gutman [35] as: where λ is a non-zero real number.

Conclusions
In this paper, we consider connected graphs of order n with vertex k-partiteness not more than m and characterize some extremal monotonic graph invariants such as the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general

Theorem 9 .
Let G be a graph of order n > 2, and the join of a complete graph with order m and a complete k-partite graph with order n − m inG n,m,k , i.e., G = K m ∨ (K k [O s 1 , • • • , O s k ]).If s 1 − 1 ≥ s 2 + 1,by moving one vertex from the part of O s 1 to the part of O s 2 , we get a new graph G

Proof.
By Lemma 9 and Theorem 9, in view of Lemmas 1 and 2, the extremal graph having the extremal value of the modified-Wiener index in G n,m,k should be the graph K m ∨ (K k [{k − t}O s , {s}O s+1 ]).