Bounds on F-index of tricyclic graphs with fixed pendant vertices

Abstract The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in Ωnα $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where Ωnα $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e 8n+12α+76≤F(G)≤8(n−1)−7α+(α+6)3 for each G∈Ωnα. $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$


Introduction and preliminaries
A representative number of a molecular graph that expresses the various features of the involved organic molecules, usually known as a topological index (TI). It plays an important role to study the certain changes in the molecular structures which may be physical or chemical. Moreover, Cheminformatics studies quantitative structural activity and property relationships that are used to examine the bioactivities and chemical reactivities of the chemical compounds in a molecular graph on the bases of obtained computational results for the di erent topological indices (TI's), see [1]. Most importantly, all the TI's are invariants under the parameter of graphs-isomorphism. For a connected graph, there are many TI's in literature. These are classi ed into three main classes degree-based TI's, distance-based TI's and polynomial-based TI's. The TI's depending upon degrees are more familiar than the others, see [2]. Wiener (1947) de ned the rst distance based TI, when he was working on para n, see [3]. Later on, it was called by Wiener index and much more work has been done on it. Recently, Furtula and Gutman (2015) [4] reinvestigated a degree-based TI and named it forgotten index (F-index). They also proposed its basic properties in the same paper and reported that it can enhance the physico-chemical capability of the molecules. The F-index and its co-index of the di erent graphs are studied by De et al. [5], Milovanovic et al. [6] and Basavanagoud et al. [7]. Khaksari and Ghorbani [8] studied the certain product of graphs with the same index. The extremal graphs with respect to F-index among the unicyclic and bicyclic graphs are studied in [9,10]. For more studies, we refer to [11] and [12][13][14][15][16][17][18][19][20][21][22][23][24][25].
In this paper, we prove the existence of extremal graphs with respect to F-index in the class of tricyclic graphs with three, four, six and seven cycles under the condition of certain pendant vertices. We also investigate the ordering and compute the bounds (lower and upper) of the F-index in the same class of graphs.
Throughout the paper, G(V(G), E(G)) for vertex-set V(G) and edge-set E(G) is considered as simple (no loops and parallel edges), nite and undirected graph. For r ∈ V(G), d(r) shows its degree (number of incident edges on r). For more theoretic terminologies, we refer [26]. Now, some important TI's are de ned as follows: De nition 1.1. For a (molecular) graph G, the rst and second Zagreb indices are De nition 1.2. For a (molecular) graph G, the general Randić index (Rα(G)) is For α = − , α = and α = , we obtain Randić, reciprocal Randić and second Zagreb indices respectively.

De nition 1.3.
For a (molecular) graph G the forgotten index (F-index) is de ned as follow: For more studies, we refer to [4,11,[27][28][29]. Following lemma is frequently used in the main results.

Computational results of F-index
A connected graph with order n and size m such that m = n − + c is called a c-cyclic graph. In particular, if c = , c = , c = or c = then it is a tree, unicyclic, bicyclic or tricyclic graph respectively. A tricyclic graph contains at least three and at most seven cycles except of exactly ve cycles. There are seven possibilities for a tricyclic graph with three cycles as shown in Figure 1. Moreover, the possibilities for the tricyclic graphs with four, six and seven cycles are four, three and one respectively, see Figure 2. Now, we de ne some more tricyclic graphs with respect to the attachment of k ≥ pendent vertices to the l vertices of the graphs which are de ned in Figure 1. To choose l vertices, we have the following choices: (i) cycle-vertex of degree 2, (ii) tree-vertex of degree 2, (iii) cycle-vertex of degree greater or equal to 2, (iv) cycle-vertex and tree-vertex of degree exactly 2, (v) cycle-vertex of degree greater or equal to 2 and tree-vertex of degree exactly 2.
More precisely, we de ne that l vertices are either of degree exactly or, greater or equal to . By joining k ≥ pendant vertices to l vertices of degree , and the vertices of degree greater or equal to of the graph G in Figure 1, the tricyclic graphs A m,r l,k, = A and A m,r l,k, = A are obtained respectively. In G , vertices of degree are four and of degree are m + m + m + r such that m = m + m + m are cycle-vertex and r are tree-vertex. Table 1 shows the vertex-partition with respect to degrees of vertices of graphs A and A .

G1
G2 G3 G4 G5 G6 G7 In G (Figure 1), the vertices of degrees , , and are , and m +m +m +r+ such that m = m +m +m are cycle-vertex and r + are tree-vertex. By joining k ≥ pendant vertices to l vertices of degree , and the vertices of degree greater or equal to of G in Figure 1, the tricyclic graphs B m,r l,k, = B and B m,r l,k, = B , are obtained, respectively. The Table 2 presents the vertex-partitions of graphs B and B .
The graph G ( Figure 1) has and m + m + m + r + vertices of degrees and , respectively such that m = m + m + m and r + are cycle-vertex. The tricyclic graphs C m,r l,k, = C and C m,r l,k, = C are obtained by joining k ≥ pendent vertices to l vertices of degree , and degree greater or equal to of the graph G in Figure 1, respectively. The Table 3 presents the vertex-partitions with respect to the degrees of vertices of the graphs C and C .
Similarly, we obtain the tricyclic graphs D m,r l,k, = D , D m,r l,k, = D , E m,r l,k, = E and E m,r l,k, = E , by joining k ≥ pendent vertices to l vertices of degree , and greater or equal to of the graphs G and G in Figure 1, respectively. In Figure 1, G has m cycle-vertex and r + tree-vertex of degrees and G has m cycle-vertex and r + tree-vertex of degrees 2. The vertex-partitions of these derived tricyclic graphs are shown in Table 4 and Table 5.
+lk +r + and (ii) the graphs G and G have the same degree sequences as of G and G respectively. For more explanation, B , B , E and E are given in Figure 2 with certain value of the parameters l, m, k and r. Now, A from A are obtained by deleting k pendant vertices from a vertex of degree k + and joining these vertices to another vertex of degree k+ . Similarly, A is derived from A by deleting k pendant vertices      Base Graphs (BG) H H H Joining k vertices to l vertices of degree = R m,r l,k, = R S m,r l,k, = S T m,r l,k, = T Joining k vertices to l vertices of degree ≥ R m,r l,k, = R S m,r l,k, = S T m,r l,k, = T Classes of tricyclic graphs generated from BG ξ ξ ξ Table 7 Base Graphs (BG) from the vertex of degree k + and joining these vertices to the vertex of degree k + . After l − iteration, we obtain A l− from A l− by deleting (l − )k pendent vertices from a vertex of degree (l − )k + and joining these vertices to the last vertex of degree k + , where ≤ l ≤ m + r. Using the same transformation, we obtain A i from A i− for ≤ i ≤ l − by the deletion of ik pendent vertices from a vertex of degree ik + and joining these vertices to the vertex of degree k + . Moreover, we obtain A i from A i− for l − ≤ i ≤ l − by the deletion of ik pendent vertices from a vertex of degree ik + and joining these vertices to the last vertex of degree Assume that U , U , U U and U , are classes of the tricyclic graphs obtained from G , G , G G and G (shown in Figure 1) respectively such that the order of each graph is m + lk + r + with lk pendent vertices. Let U lk n be a class of all the tricyclic graphs with three cycles such that each graph has order n and pendant vertices lk, where k ≥ , n ≥ and ≤ l ≤ n. Similarly, tricyclic graphs with four and six cycles obtained from the base graphs presented in Figure 2 are given in Table 6 and Table 7. Moreover, ξ lk n and ζ lk n are classes of all the tricyclic graphs with four and six cycles respectively that include each graph of order n and pendant vertices lk. Finally, we obtain the tricyclic graphs with seven cycles (K m,r l,k, = K ) and (K m,r l,k, = K ) from the base graph K (see, Figure 2) and µ lk n be a class of all the tricyclic graphs with seven cycles such that each graph has order n and pendant vertices lk. Now, by the deletion and addition of pendant vertices, we have Now, we present some important lemmas which are frequently used in the next section of main results. f (u, v) and f (u, v) are strictly decreasing functions.
By the use of De nition 1.3 and the generalization of Table 1-Table 5 for the ith iteration of the deletion of ik pendant vertices from the vertex of degree ik + and joining them to a vertex of degree k + , we obtain the F-index of the tricyclic graphs A i , B i , C i , D i and E i with three cycles and lk pendant vertices for ≤ i ≤ l − in the following lemma.  for ≤ i ≤ l − , lk + (m + r + ) + (k + ) + (k + ) + (lk − k + ) ; for i = l − , lk + (m + r + ) + (k + ) + (lk − k + ) + ; for i = l − , lk + (lk + ) + (m + r + ) + ; for i = l − , for ≤ i ≤ l − , lk + (k + ) + (m + r + ) + (lk − k + ) ; for i = l − , lk + (lk + ) + (m + r + ) + ; for i = l − , for ≤ i ≤ l − , lk + (k + ) + (lk − k + ) + (m + r + ); for i = l − , lk + (lk + ) + (m + r + ) + ; for i = l − , Proof. (a) Since the degree sequences of the base graphs of the tricyclic graph with four cycles T i j (see, H in Figure 2) and tricyclic graph with three cycles A i j (see, G in Figure 1) are equal. Therefore, the degree sequences of the graphs T i j and A i j having k ≥ pendant vertices attached with l vertices of degree (i) exactly 2 for j = and (ii) greater or equal 2 for j = are equal. Consequently, by Lemma 1.1, F(T i j ) = F(A i j ). Similarly, the degree sequences of the tricyclic graphs with four cycles S i j and R i j are equal to the degree sequences of the tricyclic graphs with three cycles B i j and D i j respectively. Thus, by Lemma 1.1, we have F(S i j ) = F(B i j ) and Proof is same as of part (a). (c) Proof is same as of part (a).

Extremal graphs with respect to F-index
The results of extremal graphs in the complete class of tricyclic graphs with xed pendant vertices are obtained in this section.  By the similar arguments as of the tricyclic graphs with three cycles, we obtain the following result for the tricyclic graphs with four, six and seven cycles.

Lower and upper bounds
The ordering and investigate of bounds (lower and upper) of the F-index in the complete class of tricyclic graphs of three, four, six or seven cycles with xed pendant vertices is given in this section. Proof.(a) Firstly, we prove that F(U ) < F(U ). For the purpose, we show that for each G ∈ U there exists G * ∈ U such that F(G) < F(G * ), where n = m + r + + lk is order of both G and G * with lk pendant vertices in each. We assume that G = A i and G * = B i for ≤ i ≤ l − . By Lemma 2.2, F(G) − F(G * ) = − < which implies that F(G) < F(G * ). Similarly, by Lemma 2.3 it can be proved that if G = A i for ≤ i ≤ l − then there exists G * = B i such that F(G) < F(G * ). In addition, if G ∈ U − {A i , A i }, using transformation of delation and joining the pendant vertices. Then, G = A i or G = A i . Thus, we get G * ∈ U such that F(G) < F(G * ). So, we conclude that F(U ) < F(U ). Similarly, we can prove that F(U ) < F(U ), F(U ) < F(U ), and F(U ) < F(U ). Consequently, we have F(U ) < F(U ) < F(U ) < F(U ) < F(U ). Proves of (b) and (c) are same as of (a).

Conclusion
In this paper, we studied the complete class of tricyclic graphs consisting on three, four, six and seven cycles for certain number of pendant vertices with respect to F-index. We proved the existence of the extremal graphs and construct the ordering of graphs with respect to F-index. Mainly, we computed the bounds (lower and upper) of F-index for the same family of graphs.