Some Entire Topological Indices of Speciﬁc Graph Families

. What distinguishes entire topological indices from other topological indices is that their formulas include information about both the edges and vertices, not just the connections between vertices. This provides more comprehensive and detailed picture of the graph’s structure. In our article, we study and analyze some entire Zagreb indices by investigating their behavior for four families of graphs; subdivision graphs, central graphs, corona products and m bridge graphs over path, cycle and complete graphs. We explore the properties of these graph structures by deriving explicit formulae for the ﬁrst, second and modiﬁed ﬁrst entire Zagreb indices for each family. Our results provide detailed information on the structural properties stored by the ﬁrst, second and modiﬁed entire Zagreb indices. These di ﬀ erent graph families show the way for future research and potential applications in ﬁelds such as chemical modeling and network investigation.


Introduction
A graph comprises both edges and a set of vertices that is not empty.In this study, we specifically examine finite graphs that are undirected, without any loops or multiple edges between the same pair of vertices.The number of vertices n and edges m in the graph G are referred to as the order and size, respectively.The degree of any vertex and edge in G is denoted by d x .The line graph L(G) is defined as the graph where each vertex represents an edge of G. Two vertices are adjacent in L(G) if and only if the corresponding edges in G share a common vertex.The path, cycle and complete graphs with n vertices are known as P n , C n and K n , respectively.For a more comprehensive understanding of the symbols and explanations, please refer to [1].
A topological index is a numerical value linked to the chemical structure, aiming to correlate the structure with various physicochemical properties, chemical reactivity, or biological activity.
In molecular modeling, these indices play a crucial role in understanding the structural features and predicting the properties or activities of molecules.
The concept of topological indices was first introduced by Harold Wiener when he discovered the initial topological index, which is called the Wiener index [2] in 1947 for searching boiling points.
One of the initial topological indices introduced is the Zagreb index, which was first introduced by Gutman and Trinajstić [3], where they investigated how the total energy of π-electron depends on the structure of molecules.The first and the second Zagreb indices for a molecular graph are defined as follows: For the latest research on Zagreb indices, we direct the reader to recent studies [4][5][6][7][8][9][10][11][12][13][14][15].
In 2018, Alwardi, A., et al. [16] introduced the definitions of the first and second entire Zagreb topological indices as shown below: x ad jacent to y or x incident to y d x d y .
In this study, we have established implicit expressions for the subdivision and central graphs pertaining to the first, second and modified first entire Zagreb indices.Recently, in [30], the same authors of this paper have introduced the modified first entire Zagreb index as x ad jacent to y or x incident to y (d x + d y ).

Entire Zagreb Indices of Some Derived Graphs
The derived graphs are those graphs which can be obtained by some particular operations from a given graph.By researching the relationships between a graph and its derived graph, one can acquire information about one based on the information on the other.In this section, we will study three types of derived graphs namely the subdivision graph, central graph and the corona product.In Figure 1, the new vertices that are added to the cycle graph are green.Proposition 2.1.For the path P n and the cycle C n , we have Proposition 2.2.Let S(K n ) be the subdivision of the complete graph K n .Then, Proof.i.There are n vertices of degree (n − 1), n(n − 1) 2 vertices of degree two and n(n − 1) edges of degree (n − 1).We have ii.For the second entire Zagreb index, we have to calculate the first part we use the partition in Table 1 and, we get Table 1.The partition of the edges in the subdivision of complete graph.
Edge type The number of edges Also, by using the adjacent edge partition as in Table 2, we have Table 2.The partition of the adjacent edges in the subdivision of complete graph.
(d e , d f ), where e f ∈ E(L(G)) Number of pairs And by using Table 3, we get iii.Similarly, as mentioned in ii, we get Proposition 2.3.Let S(K r,s ) be the subdivision of complete bipartite graph K r,s .Then iii.MM E 1 (S(K r,s )) = rs 3 + sr 3 + 4sr 2 + 8rs + 4rs 2 .
Proof.i.We have r vertices of degree s, s vertices of degree r and rs vertices of degree two.
Similarly, for the edges we have rs edges of degree s and rs edges of degree r.We get ii.For the second entire Zagreb index, we have to calculate the first part we use the partition in Table 4 and, we get iii.Similarly, as ii, we get Theorem 2.4.Let G be any graph with n vertices, m edges, and S(G) its subdivision.
x ad jacent to y or x incident to y x ad jacent to y or x incident to y  In Figure 2 the green vertices represent the new vertices in the central of path graph.
Theorem 2.5.Let C(P n ) be a central of path graph.Then Proof.i.We have n vertices of degree (n − 1) and (n − 1) vertices of degree two.
Similarly, for the edges we have 2(n − 1) edges of degree (n − 1) and (n − 1)(n − 2)/2 edges of degree (2n − 4), Thus, ii.For the second entire Zagreb index, we have to calculate the first part, we use the partition in Table 7 and, we get Table 7.The partition of the edges in the central of path graph.
Edge type The number of edges Also, by using the adjacent edge partition as in Table 8, we have According to the partition in iii.Similaly, as ii, we get Theorem 2.6.Let C(C n ) be a central of cycle graph, we have Proof.i.We have n vertices of degree (n − 1), n vertices of degree two, 2n edges of degree (n − 1) and n(n − 3) 2 of degree (2n − 4).we get, ii.For the second entire Zagreb index, we have to compute the initial segment, we utilize the division shown in the Table 10.The partition of the edges in the central of cycle graph.
Edge type The number of edges Also, by using the adjacent edge partition as in Table 11, we have And by using Table 12, we get Thus, iii.Straightforwardly, as ii, we get Theorem 2.7.Let be G any graph with n vertices, m edges and C(G) is the central graph of G. Then Proof.From the definition of the central graph, we observe that there are two types of vertices; m vertices of degree two and n vertices of degree n − 1, where n is the number of vertices in G.In the same way for edges there are two types of edges according to their degrees; n(n − 1) − 2m 2 edges of degree 2n − 4 and 2m of degree n − 1. Then x ad jacent to y or x incident to y iii.MM E 1 (G) = x ad jacent to y or x incident to y Proof.i.There are (n − 2) vertices of degree (m + 2), two vertices of degree (m + 1), 2n vertices of degree two and n(m − 2) vertices of degree three.
ii. Regarding the second entire Zagreb index, we have by utilizing the partition specified in the Table 13, we calculate the first part, Table 13.The partition of the edges in the corona product.
Edge type The number of edges Also, by using the adjacent edge partition as in Table 14, we have Table 14.The partition of the adjacent edges in the corona product.
(d e , d f ), where e f ∈ E(L(G)) Number of pairs Additionally, by utilizing Table 15, we have Table 15.The partition of the vertices incident with the edges in the corona product.E d v ,d e , where v incident to e Number of pairs Thus, iii.In the same manner , as ii, we get Theorem 2.9.[30] For any graph G with m edges, we have: Proposition 2.10.[16] For any two graphs G 1 and G 2 with Theorem 2.11.[28] Let G be a graph with n vertices and m edges.Then Theorem 2.12.For any two graphs G 1 and G Proof.By Theorem 2.9, we can write Using the results of Proposition 2.10 and Theorem 2.11, we get Theorem 2.13.Let G m be a bridge graph over path P n .Then, Proof.i.We have mn − 2m + 2 vertices of degree two, m vertices of degree one and m-2 vertices of degree three.
Similarly, for the edges we have mn − 3m + 2 edges of degree two, m of degree one, m of degree three and m − 3 of degree four, Thus, ii.For the second entire Zagreb index, we get to calculate the first part we use the partition in Table 16 and, we get And by using Thus, iii.Similarly, as ii, we get Proof.i.There are mn − m vertices of degree two, two vertices of degree three and m − 2 vertices of degree four.
In the same way, for the edges we have mn − 2m edges of degree two, four edges of degree three, 2m − 4 edges of degree four, two edges of degree five and m − 3 edges of degree six, we get ii.For the second entire Zagreb index, we get to compute the first part, we use the partition in Table 19 and, we get    Theorem 2.15.Let G m be a bridge graph over complete K n .Then, Proof.i.We have two vertices of degree n, m − 2 vertices of degree n + 1, mn − m vertices of degree n − 1.
Similarly, for the edges we have two edges of degree 2n Also, by using the adjacent edge partition as in Table 23, we have And by using Table 24, we get iii.Likewise, as mentioned in ii, we get

2. 1 .
Entire topological indices of the subdivision graph.The subdivision of a graph G, denoted by S(G), is obtained by inserting an additional vertex into each edge of G,[26].Subdivision graphs are used to obtain several mathematical and chemical properties of more complex graphs from more basic graphs.The subdivision graph of the cycle graph is illustrated in Figure1.

Figure 2 .
Figure 2. Central graph of the path C(P n ).

2. 4 .
Entire topological indices of the m bridge over graphs.A bridge graph is a graph obtained from the number of graphs G 1 , G 2 , G 3 ....G m by associating the vertices v i and v i + 1 by an edge for every i = 1, 2...m − 1, [29].The m bridge play a crucial role in network analysis and wireless communications.They aid in comprehending network connectivity, facilitating the identification of pathways in wireless communications.Moreover, they are employed to model particular network setups or evaluate the effectiveness of wireless networks.Analyzing communication and structural patterns within networks is of paramount importance.The bridge graph over path and cycle are illustrated in Figure 3, 4.

Figure 4 .
Figure 4. Bridge graph over cycle C n .

Table 3 .
The partition of the vertices incident with the edges in the subdivision of complete graph.E d v ,d e , where v incident to e Number of pairs

Table 4 .
The partition of the edges in the subdivision of complete bipartite graph .

Table 5 .
The partition of the adjacent edges in the subdivision of complete bipartite graph.= rs 3 + 2rs 2 + 2r 2 s + r 3 s.

Table 6 .
The partition of the vertices incident with the edges in complete bipartite graph.E d v ,d e , where v incident to e Number of pairs

Table 8 .
The partition of the adjacent edges in the central of path graph.
(d e , d f ), where e f ∈ E(L(G)) Number of pairs

Table 9 .
The partition of the vertices incident with the edges in the central of path graph.E d v ,d e , where v incident to e Number of pairs

Table 10
and, we get uv∈E(G)

Table 11 .
The partition of the adjacent edges in the central of cycle graph.
(d e , d f ), where e f ∈ E(L(G)) Number of pairs

Table 12 .
The partition of the vertices incident with the edges in the central of cycle graph.E d v ,d e , where v incident to e Number of pairs 2n4.The corona product of two graphs G 1 and G 2 , is a graph denoted by G 1 • G 2 which is constructed by taking |n 1 | copies of G 2 and joining each vertex of the ith copy with vertex u ∈ V(G 1 ).The corona product of two graphs can represent networks with hierarchical structures, like molecules with atoms and surrounding bonds, or transportation systems with stations and connecting routes.The corona product P n • P m of two graphs P n and P m with V(P n 2.3.Entire topological indices of the corona product of two graphs.

Table 17 .
The partition of the adjacent edges in the bridge graph over path.(d e , d f ), where e f ∈ E(L(G)) Number of pairs

Table 18 .
The partition of the vertices incident with the edges in the bridge graph over path.E d v ,d e , where v incident to e Number of pairs

Table 19 .
The partition of the edges in the bridge graph over cycle.Edge type The number of edges

Table 20 .
The partition of the adjacent edges in the bridge graph over cycle.

Table 21 .
The partition of the vertices incident with the edges in the bridge graph over cycle.E d v ,d e , where v incident to e Number of pairs

Table 22 .
Table 22 and, we getuv∈E(G) d u d v = −6n 2 + n 4 m − 3n 3 m + 9n 2 m − 5nm + 6m − 10 2. The partition of the edges in the bridge graph over complete.Edge type The number of edges