Some New Bounds of Weighted Graph Entropies with GA and Gaurava Indices Edge Weights

College of Mechanical and Electronics Engineering, Dongguan Polytechnic, Dongguan 523808, Guangdong, China Department of Mathematics and Statistics, #e University of Lahore, Lahore, Pakistan Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China


Introduction
e branch of mathematics known as graph theory provides tools for solving problems of information theory, computer sciences, physics, and chemistry [1][2][3]. Among them, special places are reserved for so-called topological descriptors, which play an important role in mathematical chemistry, especially in QSPR/QSAR surveys. Many topological descriptors are introduced and studied in the literature, such as the Zagreb index [4][5][6], the Randić connectivity index [7], and the modified Zagreb index [8,9].
In the year 2009, Vukičević and Furtula [10] introduced the geometrical arithmetic (GA) index as a molecular descriptor, and mathematical formula for this index is, ere are many interesting attributes on topological indices, and different physicochemical properties of hydrocarbons can be obtained from this index [11][12][13][14][15][16]. e predictive power of the GA index is compared with others such as the famous Randić index [17]. Due to this reason, different versions of the GA index are now investigated and introduced in the literature [18][19][20]. e other famous indices are given in [21]. e mathematical formulae for first and second Gaurava indices are respectively. Many problems of information theory, biology, computer sciences, chemistry, and discrete mathematics are directly solved by utilizing different kinds of graph measures and the graph entropy is one of the powerful tools [22][23][24][25][26] that help to understand the structural complexity of graph networks [27]. Different molecular descriptors are used to introduce weighted graph entropies [28][29][30][31].
In this paper, we extended the work of [28,30] and introduced weighted graph entropies by using GA and Gaurava indices as edge weights. We examine the extreme properties of these entropies for some special graph families. We also computed these entropies for different chemical structures. Now, we define some basic notions about entropy of graphs. We always consider G being a connected graph with E as the set of edges, V as the set of vertices, and w to be the edge weight given to the edges of graph G that will be used to define weighted graph entropy. As mentioned above, GA and Gaurava indices are taken as edge weights in this paper. d u denoted the degree of a vertex u that is defined as the total number of vertices of G that are at distance one from the vertex u. Consider where w(v i v j is the weight v i v j and w(v i v j ) > 0. Now, the weighted graph entropy can be defined by Definition 1. For the graph G, the weighted entropy can be defined as follows [28,30]: Here, p uv is same as that given in (4).

Main Results
In this section, we are going to present our main results.

Lemma 1.
Let G be a simple graph having n vertices and m edges and let Δ and δ be the maximum degree and minimum degree of a vertex, respectively; then, with equality if and only if G is a regular graph or G is a bipartite graph.

Lemma 2.
Consider G is a simple connected graph having m edges. en, we have with equality if and only if G is isomorphic to K 1,n− 1 or G is isomorphic to regular graph or G is isomorphic to (Δ, 1) semiregular graph.

Lemma 3.
Consider G is a simple connected graph having m edges. en, we have Moreover, the equality holds if and only if G is isomorphic to a regular graph or G is isomorphic to (Δ, 1) semiregular graph.

Lemma 4. Consider G is a simple connected graph having m edges. en, we have
with equality if and only if G is isomorphic to K 1,n− 1 or G is isomorphic to a complete graph K 3 .

Lemma 5.
Consider G is a simple connected graph having m edges. en, we have Theorem 1. Consider G is a connected graph having n vertices with n ≥ 3. en, we have Proof. We prove the result for GO 1 , the other results can be proved in the same manner.
Since G is a connected graph with n number of vertices, for any vertex, the maximum possible degree can be n − 1 and the minimum possible degree can be one. Hence, for any edge uv, the minimum degrees for u and v can be 1 and 2 and maximum possible degrees for u and v can be n − 1 and n − 1, and hence, we have erefore, □ Theorem 2. Let G be a graph with n vertices. Let δ and Δ be the minimum and maximum degrees of G, respectively. en, we have Proof. Since G is the connected graph with n number of vertices, for any vertex, the maximum possible degree can be n − 1 and the minimum possible degree can be one. Hence, for any edge uv, the minimum degrees for u and v can be 1 and 2 and maximum possible degrees for u and v can be n − 1 and n − 1, and hence, we have Also, □ Mathematical Problems in Engineering Theorem 3. Consider G is a regular graph having n vertices with n≽3. en, we have Note that left inequality turns into equality if G is a cyclic graph, and the right inequality turns into equality if G is a complete graph.

Note that the left inequality turns into equality if G is a star graph, and log(n − 1) � log(⌊n/2⌋⌈n/2⌉) if and only if G is a complete bipartite graph (balanced).
Theorem 5. Let T be a tree of order n(n > 2) with maximum degree vertex Δ; then, we have Theorem 6. Let G be a simple graph having n vertices and m edges and let Δ and δ be the maximum degree and minimum degree of a vertex, respectively; then, with equality if and only if G is either a regular graph or a bipartite graph.
In graph theory, the molecular graph is obtained by taking atoms as vertices and bounds as edges. It can be noted that the maximum possible degree for a vertex in a molecular graph is four. Following theorem is about the bounds of weighted entropy for the molecular graph.

Relation of Entropy with Zagreb Indices
Theorem 8. Consider G is a simple connected graph. en, we have e equality holds if and only if G is a union of K 2 .
Theorem 9. Consider G is any graph, then we have Equality holds if and only if G is isomorphic to the regular graph.

Theorem 10. Consider G is any graph, then we have
Equality holds if and only if G is isomorphic to the regular graph.

Theorem 11. Consider G is any graph, then we have
Equality holds if and only if G is isomorphic to the regular graph.

Numerical Examples.
Here, we compute the weighted entropies introduced in this paper for some chemical structures.

Example 1.
Consider the porphyrin dendrimers shown in Figure 1. We denote the graph of porphyrin dendrimers by G, and the edge partition of G is given in Table 1. Using Table 1 and definition of entropy, we have the following entropies for porphyrin dendrimers: , Example 2. e graph G of zinc-porphyrin dendrimer is shown in Figure 2, and the edge partition for this dendrimer is given in Table 2. We have the following computations for the entropies of zinc-porphyrin dendrimer.     Example 3. Let G be the graph of poly(ethylene amidoamine) dendrimers as shown in Figure 3. en, the edge partition of this dendrimer is given in Table 3 and we have the following results.

Conclusion
In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. is measure, first introduced by Körner in the 1970s, has since also proven itself useful in other settings, including combinatorics. In this paper, we have studied graph entropy with GA and Gaurava indices and justified it by some numerical examples. It would be interesting to work on entropy of weighted graphs with some other degree-and distance-based topological indices. e bounds of degree-based network entropy can also be used in national security, Internet networks, social networks, structural chemistry, ecological networks, computational systems biology, etc. ey will play an important role in analyzing structural symmetry and asymmetry in real networks in the future.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they do not have any conflicts of interest.