On bounds for topological descriptors of φ-sum graphs

The properties of chemical compounds are very important for the studies of the non-isomorphism phenomenon's related to the molecular graphs. Topological indices (TIs) are one of the mathematical tools which are used to study these properties. Gutman and Trinajsti [Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem Phys Lett. 1972;17(4):535–538] defined the Zagreb indices (descriptors) to find correlation value between a molecular graph and its total π-electron energy. Later on, Bollobás and Erdös [Graphs of extremal weights. Ars Comb; 1998;50:225–233] defined the most general form of these indices (descriptors) called by general Randić index (GRI) and first general Zagreb index (FGZI), respectively. In this paper, we computed the bounds for FGZI and GRI of φ-sum graphs, obtained by the strong product of the graph with another graph Γ, where is constructed using four subdivision operations on the graph G. At the end, we also include the results for some particular families of graphs as the applications of the obtained results.


Introduction
A number, polynomial or a matrix can uniquely identify a graph and a topological index (TI) of a molecular graph is a numeric number that can be defined as a function F : aCa −→ aR, where C is a class of molecular graphs and R is a set of real numbers. TI's are classified into different classes but degree-based are most familiar, see [1]. These are used to characterize the physicochemical properties of chemical compounds of the molecular graphs like surface tension & density, melting & freezing point, heat of evaporation & formation and solubility, see [2][3][4]. TI's are also used in the studies of the structural properties of computer-based networks such as clustering, connectivity, modularity, robustness and vulnerability, see [5].
In computational graph theory, the concept of formation of the new graphs by using some operations is studied widely. For a connected molecular graph G, Yan et al. [6] defined the new graphs called as line graph L(G), subdivided graph S(G), triangle parallel graph R(G), line superposition graph Q(G) and total graph T(G) using the subdivision related operations L, S, R, Q and T on G, respectively. They also obtained the Wiener index of these new resultant graphs φ(G), where φ ∈ {L, S, R, Q, T}. Eliasi et al. [7] defined the φ-sum graphs (G φ ) using the operation of cartesian product on the graphs φ(G) and . They also obtained the Wiener index of these φ-sum graphs G S , G R , G Q and G T . The first, second and forgotten zagreb indices of the φ-sum graphs are computed in [8,9]. The first general Zagreb index and general sum-connectivity index of the aforesaid cartesian product-based φ-sum graphs are obtained in the form of exact formulas and bounds, see [10][11][12]. For further studies of the TIs on the graphs obtained by the various operations of graphs, we refer to [13][14][15][16][17][18][19][20][21].
Recently, Sarala et al. [22] obtained the F-index of the strong product based φ-sum graphs. The theme of this note is to compute FGZI and GRI of φ-sum graphs G +φ constructed by the strong product of graphs φ(G) and , where φ(G) ∈ {(S(G), R(G), Q(G), T(G)}. The remaining paper is organized as: Section 2 contains some basic definitions and operations on graphs G and φ(G), Section 3 covers the main results of upper and lower bounds of FGZI and GRI and Section 4 is devoted to conclusion.

Preliminaries
Let G = (V(G), E(G)) be an undirected, simple, finite and connected molecular graph with vertex set V(G) and edge set E(G) such that each vertex presents atom and each edge shows the bonding among the atoms. Degree of u in G is G (u) = |{x ∈ V(G) : d(x, u) = 1}|. The maximum and minimum degrees of a graph G are defined as G = max{ G (u) : ∀ u ∈ V(G)} and δ G = min{ G (u) : ∀ u ∈ V(G)}. We note that δ G ≤ G (u) ≤ G , where equality holds if and only if G is a regular graph. For further study of graph-theocratic terminologies, see [23][24][25]. Now, we define some important degree-based TIs. Definition 2.1: For a molecular graph G, the first and second Zagreb indices (descriptors) are defined as: In 1972, Gutman and Trinajsti defined these indices (descriptors) to study the molecular graphs, see [26][27][28].

Operations of Subdivision:
The following operations are defined in [7].   graph G +φ based on strong product of graphs φ(G) and is a graph with vertex set Figure 2.

Main results
This section contains the main results of φ-sum graphs based on strong product. Assume that the connected graphs G and have number of vertices n G and n , and number of edges m G and m , respectively. (a) We have α 2 ≤ M α (G +s ) ≤ α 1 , where α 1 , α 2 ≥ 0 and

Proof: By definition
In terms of edges, where n = α − 1 Consequently, Similarly, we can compute equality holds iff G and are regular graphs. Therefore, Similarly, we can compute equality holds iff G and are regular graphs.

Theorem 3.2:
Let G and be two connected graph.
Proof: By definition Hence equality holds iff G and are regular graphs. Consequently, Similarly, we can compute equality holds iff G and are regular graphs.

Proof: By definition
Since s 1 is vertex inserted in the edge w i w j of G and s 2 is vertex inserted in the edge w j w k of G for all w i , w j w k V(G), we have Similarly, we can compute equality holds iff G and are regular graphs. Consequently, Similarly, we can compute equality holds iff G and are regular graphs.

Theorem 3.4:
Let G and be two connected graph.
equality holds iff G and are regular graphs.

Applications and discussion
The upper and lower bounds on the FGZI and GRI of P n+φ P m for α, β > 0 are given as follows: For S-sum graph P n+S P m : For R-sum graph P n+R P m : For Q-sum graph P n+Q P m : For T-sum graph P n+T P m : (a) α 1 = 6(12) α−1 (n − 1)(2m − 1) α 2 = 6(4) α−1 (n − 1)(2m − 1) Now, we present the numerical values of FGZI and GRI for P n+φ P m with the help of the above bounds under the assumption that n = m = α = β and α, β > 0 in Tables 1-4. Moreover, Tables 5 and 6 present the exact values of FGZI and GRI for the same graphs (Figures 3-6).

Conclusion
In this paper, we have computed the bounds (upper and lower) for FGZI and GRI of φ-sum graphs which are obtained by the strong product of φ(G) graph with another graph , where φ(G) is constructed using four subdivision operations on the graph G. At the end, we also included the results for some particular families of graphs as the applications of the obtained results and concluded that the exact values satisfy the obtained bounds. More preciously, upper and lower bounds for FGZI and GRI of φ-sum graphs based on strong product are computed, where α, β > 0. In addition, if we assume that α, β < 0 then these bounds become α 1 ≤ M α (G +φ ) ≤ α 2 and β 1 ≤ R β (G +φ ) ≤ β 2 .