On Computing Degree-Based Topological Invariants for Special Graphs

The origin of graph theory was traditionally dated back to year 1736, when Euler solved the famous Königsberg bridges problem [1]. Graph theory is one of the few mathematical areas of study that have an exact date of emergence. Kirchhoff's research into electrical networks directed his progress of basic theory as well as ideas about trees in graphs, while Cayley considered trees from the calculation of isomers as living chemicals. Another form of graph paradox was suggested by Hamilton. Graph theory, a branch of mathematics relating to network points connected by lines, has its origin in recreational mathematical problems but has grown into the core of mathematical research through chemistry, occupational research, social science, and computer science. A graph \(G\) consists of a set of nodes \(V=\{u_1,u_2,u_3,\ldots\}\) and the other set \(E=\{e_{i_1},e_{i_2},e_{i_3},\ldots\}\) whose elements are called arcs. If \(e_i=u_1 u_2\) is an arc of graph \(G\), then the nodes \(u_1\) and \(u_2\) are adjacent in \(G\). When any two nodes are connected by more than one arc, the graph is called a multigraph. A numerical value associated with each node is its degree.

The topological descriptor is the nominal value associated with the graph and must be a structural variable. Several topological descriptors have been described and most of them have found properties as a modeling of chemicals, drugs, and other molecular structures. The Wiener index [2], [3], [4] is the first topological descriptor used in chemistry. Topological descriptors are statistical measures of a graph of cells that remain constant under the graph isomorphism. Started their journey in 1947, a wide variety of topological descriptors have been created and much work has been done on graphs of various cells and networks. Likewise, Harold Wiener introduced and developed this intriguing guide to determine the physical characteristics of alkane species known as paraffin in 1947. Topological descriptors are of incompatible types as degree-based topological descriptors, distance-based topological descriptors, and spectrum-based [5], [6] topological descriptors.

To adopt an applied mathematical approach, graphs like web, windmill, and Dutch windmill graphs are very useful and idealized models of networks having a central or a well-connected center along with peripheral or repeating components. These graphs can be used to model connectivity patterns in case of molecular and chemical networks, communication or transport networks, and many other types of complex network models that include interaction between a few well-connected units and many other less connected units. In the case of the chemical graph model, such a type of family of graphs is quite suitable to depict simple models of molecular frameworks involving branchings and repeating parts of molecules along with the concept of hubs. The use of double and strong double construction allows us to investigate the effect of increasing connectivity and coupling between the parts of graphs [7], [8]. In this paper, the Sombor indices calculated for these graph models considered the degree of vertices as a measure of structural irregularity, and thus the comparison of these indices could be helpful in distinguishing the features of these families of graphs.

This paper discussed simple graphs, which are graphs with no multiple loops and arcs. There are some important degree-based topological descriptors defined [4], [5]. The first and second Zagreb descriptors [9] are:

In this paper \(d_G(u_1)\) and \(d_G(u_2)\) represent the degree of the vertices \(u_1\) and \(u_2\) respectively.

Suppose \(G\) show a molecular structure, then the first general Zagreb descriptor [10] was defined as:

where, $\beta \neq 0$, $\beta \neq 1$ and $\beta \in \mathbb{R}$. The first degree-based invariant was defined as:

Fajtlowicz explained the Harmonic indicator in 1988 [11] and it was defined as:

The inverse sum indicator [12] is stated as:

Symmetric division indicator [13] is defined as:

The geometric arithmetic indicator was suggested by Vukičević and Furtula [14].

In the following section, the degree-based Sombor descriptors are calculated. These Sombor descriptors [15], [16], [17], [18], [19] are defined as:

We discussed web, windmill, Dutch windmill, double graph, and strong double graph of Dutch windmill [20], [21], [22] with the assistance of Sombor index. The chemical property of the Sombor indicator, mainly the capability to discriminate and speculate, was explored by Gutman et al. [23] and Shigehalli and Chougale [24]; obviously the Sombor index has gained considerable awareness in chemistry and mathematics. In this research work, the Sombor indices of five particular families of graphs have been studied, starting from the process of formation of the graph family to degree study, mathematical calculation, numerical computation, and comparison. In the current scenario \(p\), represents the size or order of the complete graph component (the number of vertices in each complete subgraph), and \(r\) is the number of complete graph components used in the formation of the graph family. There are three Sombor-type parameters considered in the current study: Sombor index \(SO\), reduced Sombor index \(SO_{\mathrm{red}}\), and average Sombor index \(SO_{\mathrm{avg}}\). It has been found that \(SO\) takes the maximum values while \(SO_{\mathrm{avg}}\) takes the minimum values. The complete methodology of this study is illustrated in Figure 1 below.

The web graph \(W_{p,r}\) was obtained by taking \(r\) copies of the cycle graph \(C_p\) as described in Figure 2. A web graph has been defined equivalently as a prism graph. The order of the web graph is \(pr\) and size \(p(2r-1)\). In the web graph, we have the following sets of edge partition:

\[ \begin{gathered} E_1=\left\{u_i u_j \in E(W_{p,r}) \mid d_{W_{p,r}}(u_i)=d_{W_{p,r}}(u_j)=3\right\},\\ E_2=\left\{u_i u_j \in E(W_{p,r}) \mid d_{W_{p,r}}(u_i)=3,\ d_{W_{p,r}}(u_j)=4\right\},\\ E_3=\left\{u_i u_j \in E(W_{p,r}) \mid d_{W_{p,r}}(u_i)=d_{W_{p,r}}(u_j)=4\right\} \end{gathered} \]

For \(p=4\) and \(r=3\), we computed non-identical Sombor indices of \(W_{4,3}\):

\[ \begin{aligned} S O\left[W_{4,3}\right] & =40+40 \sqrt{2}, \\ S O_{r e d}\left[W_{4,3}\right] & =8 \sqrt{13}+28 \sqrt{2}, \\ S O_{a v g}\left[W_{4,3}\right] & =\frac{8 \sqrt{65}+60 \sqrt{2}}{3} \end{aligned} \]

The final description of the web graph for non-identical Sombor descriptors are given in theorem (1), (2), and (3).

Theorem 1. Suppose \(W_{p,r}\) represent a web graph and \(O|W_{p,r}|=pr\). Then, the \(SO\) of \(W_{p,r}\) is

Proof. For \(p \geq 3\) and \(r \geq 3\), consider \(W_{p,r}\) as a web graph with \(O|W_{p,r}|=pr\). We know,

\[ \begin{gathered} SO\left(W_{p,r}\right)=\sum_{u_i u_j \in E\left(W_{p,r}\right)} \sqrt{d_G(u_i)^2+d_G(u_j)^2},\\ SO\left(W_{p,r}\right)=2p\sqrt{18}+10p+p(2r-5)\sqrt{32} \end{gathered} \]

Theorem 2. Let \(W_{p,r}\) represent a web graph and \(O|W_{p,r}|=pr\). Then, the \(SO_{\mathrm{red}}\) of \(W_{p,r}\) is

Proof. For \(p \geq 3\) and \(r \geq 3\), consider \(W_{p,r}\) as a web graph with \(O|W_{p,r}|=pr\). We know,

$ \begin{gathered} SO_{\mathrm{red}}\left(W_{p,r}\right)=\sum_{u_i u_j \in E(W_{p,r})} \sqrt{\left(d_G(u_i)-1\right)^2+\left(d_G(u_j)-1\right)^2},\\ SO_{\mathrm{red}}\left(W_{p,r}\right)=2p\sqrt{8}+2p\sqrt{13}+p(2r-5)\sqrt{18} \end{gathered} $

Theorem 3. Suppose \(W_{p,r}\) represent a web graph and \(O|W_{p,r}|=pr\). Then, the \(SO_{\mathrm{avg}}\) of \(W_{p,r}\) is

Proof. For \(p \geq 3\) and \(r \geq 3\), consider \(W_{p,r}\) as a web graph with \(O|W_{p,r}|=pr\). We know,

$SO_{\mathrm{avg}}\left(W_{p,r}\right)=\sum_{u_i u_j \in E(W_{p,r})} \sqrt{\left(d_G(u_i)-\frac{2m}{n}\right)^2+\left(d_G(u_j)-\frac{2m}{n}\right)^2} $

Therefore, it necessitates the outcome.

After having outcomes from theorem (1), (2), and (3), we computed the relationship [18] among the results of \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\) of a web graph. Figure 3 and Table 1 define the graphic and numeric forms of these descriptors.

Table 1 reveals the consistent increase of the three Sombor indices. For every value considered, the indices satisfy:

$ SO(W_{p,r})>SO_{\mathrm{red}}(W_{p,r})>SO_{\mathrm{avg}}(W_{p,r}) $

The growth is substantial: from \((p,r)=(3,3)\) to \((10,10)\), \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\) increased by factors of approximately \(14.27\), \(14.90\), and \(16.05\), respectively. Moreover, at \((10,10)\), \(SO_{\mathrm{red}}/SO\approx 0.740\) and \(SO_{\mathrm{avg}}/SO\approx 0.492\). Thus, although all three indices exhibit similar increasing behavior, \(SO_{\mathrm{avg}}\) shows the comparatively stronger relative growth over this range.

The windmill graph \(W_p^r\) was constructed by using \(r\) copies of the complete graph with a common node as shown in Figure 4. The windmill graph [23] is isomorphic to \(rK_{p-1}+K_1\). The \(O|W_p^r|\) is \(r(p-1)+1\), and size \(\frac{rp(p-1)}{2}\). In the windmill graph, we have the following sets of edge partition:

$ \begin{gathered} E_1=\left\{u_i u_j \in E(W_p^r) \mid d_{W_p^r}(u_i)=d_{W_p^r}(u_j)=p-1\right\},\\ E_2=\left\{u_i u_j \in E(W_p^r) \mid d_{W_p^r}(u_i)=p-1,\ d_{W_p^r}(u_j)=r(p-1)\right\} \end{gathered} $

For \(p=3\) and \(r=3\), we computed non-identical Sombor indices of \([W_3^3]\):

$ \begin{gathered} SO[W_3^3]=12\sqrt{10}+6\sqrt{2},\\ SO_{\mathrm{red}}[W_3^3]=6\sqrt{26}+3\sqrt{2},\\ SO_{\mathrm{avg}}[W_3^3]=\frac{24\sqrt{37}}{7}+\frac{12\sqrt{2}}{7} \end{gathered} $

The final description of the windmill graph for non-identical Sombor descriptors were given in theorem (4), (5), and (6).

Theorem 4. Let \(W_p^r\) represent a windmill graph and \(O|W_p^r|=2p-1\). Then, the \(SO\) of \(W_p^r\) is

Proof. By letting \(W_p^r\) as a windmill graph and \(O|W_p^r|=2p-1\), where \(r\geq 2\) and \(p\geq 3\). We know,

$ \begin{gathered} SO(W_p^r)=\sum_{u_i u_j \in E(W_p^r)} \sqrt{d_G(u_i)^2+d_G(u_j)^2},\\ SO(W_p^r)=r(p-1)\sqrt{(p-1)^2+(r(p-1))^2}+\frac{r\sqrt{2}}{2}(p-1)^2(p-2) \end{gathered} $

Theorem 5. Suppose \(W_p^r\) is a windmill graph with \(O|W_p^r|=2p-1\). Then,

Proof. By letting \(W_p^r\) as a windmill graph and \(O|W_p^r|=2p-1\), where \(r\geq 2\) and \(p\geq 3\). We know,

$ \begin{gathered} SO_{\mathrm{red}}(W_p^r)=\sum_{u_i u_j \in E(W_p^r)} \sqrt{(d_G(u_i)-1)^2+(d_G(u_j)-1)^2},\\ SO_{\mathrm{red}}(W_p^r)=r(p-1)\sqrt{(p-2)^2+((p-1)(p-1)+(p-2))^2}+\frac{r}{2}\sqrt{2}(p-1)(p-2)^2. \end{gathered} $

Theorem 6. Let \(W_p^r\) be a windmill graph and \(O|W_p^r|=2p-1\). Then, the \(SO_{\mathrm{avg}}\) of \(W_p^r\) is

Proof. By letting \(W_p^r\) as a windmill graph and \(O|W_p^r|=2p-1\), where \(r\geq 2\) and \(p\geq 3\). We know,

$SO_{\mathrm{avg}}(W_p^r)=\sum_{u_i u_j \in E(W_p^r)} \sqrt{\left(d_G(u_i)-\frac{2m}{n}\right)^2+\left(d_G(u_j)-\frac{2m}{n}\right)^2} $

Therefore, it necessitates the outcome.

After getting values from theorem (4), (5), and (6), we computed the relationship [18] between the results of Sombor descriptors of a windmill graph. The graphical and numerical comparison of these descriptors are available in Figure 5 and Table 2.

Table 2 indicates a pronounced increase in all three Sombor indices. The ordering remains unchanged throughout the table:

$ SO(W_p^r)>SO_{\mathrm{red}}(W_p^r)>SO_{\mathrm{avg}}(W_p^r) $

The increase is considerably faster than that observed for the web graph. From \((p,r)=(3,3)\) to \((10,10)\), \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\) increased by factors of approximately \(274.00\), \(347.77\), and \(329.19\), respectively. At \((10,10)\), the ratios \(SO_{\mathrm{red}}/SO\) and \(SO_{\mathrm{avg}}/SO\) were approximately \(0.952\) and \(0.602\), respectively. These results demonstrated the rapid growth of the indices for the windmill graph as both parameters increased.

Dutch windmill graph was constructed by taking \(r\) copies of the \(C_p\) where, \(p\geq 3\) joined by one common node as demonstrated in Figure 6. It was symbolized by \(D_p^r\), where \(p\geq 3\) and \(r\geq 2\). The size of the Dutch windmill graph [23] is \(pr\) and \(O|D_p^r|=r(p-1)+1\). In the Dutch windmill graph, we have the following sets of edge partition:

$ \begin{gathered} E_1=\left\{u_i u_j \in E(D_p^r) \mid d_{D_p^r}(u_i)=d_{D_p^r}(u_j)=p-1\right\},\\ E_2=\left\{u_i u_j \in E(D_p^r) \mid d_{D_p^r}(u_i)=p-1,\ d_{D_p^r}(u_j)=2r\right\} \end{gathered} $

For \(p=3\) and \(r=2\), we computed non-identical Sombor indices of \([D_3^2]\):

$ \begin{gathered} SO[(D_3^2)]=8\sqrt{5}+4\sqrt{2},\\ SO_{\mathrm{red}}[(D_3^2)]=4\sqrt{10}+2\sqrt{2},\\ SO_{\mathrm{avg}}[(D_3^2)]=\frac{8\sqrt{17}}{5}+\frac{4\sqrt{2}}{5} \end{gathered} $

The final description of the Dutch windmill graph for Sombor descriptors is available in theorem (7), (8), and (9).

Theorem 7. Let \(D_p^r\) be a Dutch windmill graph and \(O|D_p^r|=r(p-1)+1\). Then, the \(SO\) of \(D_p^r\) is

Proof. By letting \(D_p^r\) as a Dutch windmill graph and \(O|D_p^r|=r(p-1)+1\), where \(p\geq 3\) and \(r\geq 2\). We know,

$ \begin{gathered} SO(D_p^r)=\sum_{u_i u_j \in E(D_p^r)} \sqrt{d_G(u_i)^2+d_G(u_j)^2},\\ SO(D_p^r)=r(p-2)\sqrt{8}+2r\sqrt{2^2+(2r)^2} \end{gathered} $

Theorem 8. Suppose \(D_p^r\) is a Dutch windmill graph and \(O|D_p^r|=r(p-1)+1\). Then, the \(SO_{\mathrm{red}}\) of \(D_p^r\) is

Proof. By letting \(D_p^r\) be a Dutch windmill graph and \(O|D_p^r|=r(p-1)+1\), where \(p\geq 3\) and \(r\geq 2\). We know,

$ \begin{gathered} SO_{\mathrm{red}}(D_p^r)=\sum_{u_i u_j \in E(D_p^r)} \sqrt{(d_G(u_i)-1)^2+(d_G(u_j)-1)^2},\\ SO_{\mathrm{red}}(D_p^r)=r(p-2)\sqrt{2}+2r\sqrt{1+(2r-1)^2} \end{gathered} $

Theorem 9. Suppose \(D_p^r\) is a Dutch windmill graph and \(O|D_p^r|=r(p-1)+1\). Then, the \(SO_{\mathrm{avg}}\) of \(D_p^r\) is

Proof. By letting \(D_p^r\) be a Dutch windmill graph and \(O|D_p^r|=r(p-1)+1\), where \(p\geq 3\) and \(r\geq 2\). We know,

$ SO_{\mathrm{avg}}(D_p^r)=\sum_{u_i u_j \in E(D_p^r)} \sqrt{\left(d_G(u_i)-\frac{2m}{n}\right)^2+\left(d_G(u_j)-\frac{2m}{n}\right)^2} $

$ SO_{\mathrm{avg}}(D_p^r)=r(p-2)\sqrt{2}\left(2-\frac{2pr}{r(p-1)+1}\right)+2r\sqrt{\left(2-\frac{2pr}{r(p-1)+1}\right)^2+\left(2r-\frac{2pr}{r(p-1)+1}\right)} $

By collecting results from theorem (7), (8), and (9), we computed the relationship [18] between the results of Sombor descriptors of a Dutch Windmill graph. The graphic and numeric form of these descriptors are defined in Table 3 and Figure 7.

Table 3 demonstrates that all three indices increased monotonically as \((p=r)\) from \(3\) to \(10\), while maintaining the ordering

$ SO(D_p^r)>SO_{\mathrm{red}}(D_p^r)>SO_{\mathrm{avg}}(D_p^r) $

However, the relative separation between the indices was more noticeable for \(SO_{\mathrm{avg}}\). From \((p,r)=(3,3)\) to \((10,10)\), \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\) increased by factors of approximately \(13.53\), \(14.17\), and \(7.6\), respectively. At \((10,10)\), \(SO_{\mathrm{red}}/SO\approx 0.786\), whereas \(SO_{\mathrm{avg}}/SO\approx 0.170\). Hence, \(SO_{\mathrm{avg}}\) remains substantially smaller than the other two indices for the \(D_p^r\).

The double graph of \(D_p^r\) (\(G\)) was represented by \(D[G]\). In this case, two copies of \(G\) would be taken and the nodes of one copy would be connected to their respective neighbors in another copy in order to make the double graph of Dutch windmill as described in Figure 8. The order of double graph of Dutch windmill graph [23] was \(2r(p-1)+2\) and size \(r(5p-3)\). It was represented by \(D[D_p^r]\). In double graph of Dutch windmill graph, we have the following sets of edge partition:

$ \begin{gathered} E_1=\left\{u_i u_j \in E(D(D_p^r)) \mid d_{D(D_p^r)}(u_i)=d_{D(D_p^r)}(u_j)=4\right\},\\ E_2=\left\{u_i u_j \in E(D(D_p^r)) \mid d_{D(D_p^r)}(u_i)=4,\ d_{D(D_p^r)}(u_j)=3p+r\right\} \end{gathered} $

For \(p=3\) and \(r=2\), we computed non-identical Sombor indices of \(D[D_3^2]\):

$ \begin{gathered} SO[D(D_3^2)]=64\sqrt{5}+32\sqrt{2},\\ SO_{\mathrm{red}}[D(D_3^2)]=16\sqrt{58}+24\sqrt{2},\\ SO_{\mathrm{avg}}[D(D_3^2)]=-\frac{32}{5}+\frac{64\sqrt{17}}{5} \end{gathered} $

The final description of \(D(D_p^r)\) for Sombor indices were available in theorem (10), (11) and (12).

Theorem 10. Consider a graph \(D(D_p^r)\) and \(O|D(D_p^r)|=2r(p-1)+2\). Then, the \(SO\) of \(D(D_p^r)\) is

Proof. By letting \(D(D_p^r)\) be a double graph of Dutch windmill graph and \(O|D(D_p^r)|=2r(p-1)+2\), where \(p\geq 3\) and \(r\geq 2\). We know,

$ \begin{gathered} SO(D(D_p^r))=\sum_{u_i u_j \in E(D(D_p^r))} \sqrt{d_G(u_i)^2+d_G(u_j)^2},\\ SO(D(D_p^r))=r(3p-5)\sqrt{32}+2r(p+1)\sqrt{(r(p+1))^2+4^2} \end{gathered} $

Theorem 11. Let \(D(D_p^r)\) be a double graph of Dutch windmill graph and \(O|D(D_p^r)|=2r(p-1)+2\). Then, the \(SO_{\mathrm{red}}\) of \(D(D_p^r)\) is

Proof. By letting \(D(D_p^r)\) be a double graph of Dutch windmill graph and \(O|D(D_p^r)|=2r(p-1)+2\), where \(p\geq 3\) and \(r\geq 2\). We know,

$ \begin{gathered} SO_{\mathrm{red}}(D(D_p^r))=\sum_{u_i u_j \in E(D(D_p^r))} \sqrt{(d_G(u_i)-1)^2+(d_G(u_j)-1)^2},\\ SO_{\mathrm{red}}(D(D_p^r))=r(3p-5)\sqrt{18}+2r(p+1)\sqrt{(r(p+1)-1)^2+9} \end{gathered} $

Theorem 12. Suppose \(D(D_p^r)\) is a double graph of Dutch windmill graph and \(O|D(D_p^r)|=2r(p-1)+2\). Then, the \(SO_{\mathrm{avg}}\) of \(D(D_p^r)\) is

Proof. By letting \(D(D_p^r)\) be a double graph of Dutch windmill graph, where \(p\geq 3\) and \(r\geq 2\). We know,

$SO_{\mathrm{avg}}(D(D_p^r))=\sum_{u_i u_j \in E(D(D_p^r))} \sqrt{\left(d_G(u_i)-\frac{2m}{n}\right)^2+\left(d_G(u_j)-\frac{2m}{n}\right)^2} $

Therefore, it necessitates the outcome.

After obtaining general results from theorem (10), (11), and (12), we computed the relationship between the results of Sombor descriptors of a double Dutch windmill graph. The graphic and numeric comparison of indices are available in Table 4 and Figure 9.

Table 4 shows a substantial increase in all three indices as \((p=r)\) increased from \(3\) to \(10\). The ordering

$ SO(D(D_p^r))>SO_{\mathrm{red}}(D(D_p^r))>SO_{\mathrm{avg}}(D(D_p^r)) $

was preserved for all considered cases. From \((p,r)=(3,3)\) to \((10,10)\), the values of \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\) increased by factors of approximately \(69.00\), \(77.18\), and \(126.06\), respectively. Interestingly, the three indices became relatively closer as the graph grew; for example, at \((10,10)\), \(SO_{\mathrm{red}}/SO\approx 0.977\) and \(SO_{\mathrm{avg}}/SO\approx 0.916\). Compared with the original Dutch windmill graph, the double construction therefore produced substantially larger index values and noticeably changed their relative growth, particularly for \(SO_{\mathrm{avg}}\).

The strong \(D[G]\) was derived from considering both graphs such that the closed adjacent of each vertex in one graph was paired with the adjacent vertex in another graph. The strong double graph [23] of a Dutch windmill graph is represented by \(SD[D_p^r]\) as shown in Figure 10. The order of a \(SD(D_p^r)\) was \(2r(p-1)+2\) and size \(r(p-1)(p+2)+(r(p+1)+1)\). In strong double graph of Dutch windmill graph, we have the following sets of edge partition:

$ \begin{gathered} E_1=\left\{u_i u_j \in E(SD(D_p^r)) \mid d_{SD(D_p^r)}(u_i)=d_{SD(D_p^r)}(u_j)=p+2\right\},\\ E_2=\left\{u_i u_j \in E(SD(D_p^r)) \mid d_{SD(D_p^r)}(u_i)=p+2,\ d_{SD(D_p^r)}(u_j)=4p+r+1\right\} \end{gathered} $

For \(p=3\) and \(r=2\), we computed three different Sombor descriptors of \(SD[D_3^2]\):

$ \begin{gathered} SO[SD(D_3^2)]=12\sqrt{2}+9\sqrt{212}\\ SO_{\mathrm{red}}[SD(D_3^2)]=12\sqrt{2}+32\sqrt{10}\\ SO_{\mathrm{avg}}[SD(D_3^2)]=13\sqrt{2}+\frac{142}{9} \end{gathered} $

The final representation of the \(SD(D_p^r)\) for non-identical Sombor indices are available in theorem (13), (14), and (15).

Theorem 13. Consider a graph \(SD(D_p^r)\) and \(O|SD(D_p^r)|=2r(p-1)+2\). Then, the \(SO\) of \(SD(D_p^r)\) is

Proof. By letting \(SD(D_p^r)\) be a strong \(D(D_p^r)\) and \(O|SD(D_p^r)|=2r(p-1)+2\), where \(p\geq 3\) and \(r\geq 2\). We know,

$ \begin{gathered} SO(SD(D_p^r))=\sum_{u_i u_j \in E(SD(D_p^r))} \sqrt{d_G(u_i)^2+d_G(u_j)^2},\\ SO(SD(D_p^r))=r(p^2-3)\sqrt{2(p+2)^2}+\sqrt{2(r(p+1)+1)^2}\\ +2r(p+1)\sqrt{(r(p+1)+1)^2+(p+2)^2} \end{gathered} $

Theorem 14. Consider a graph \(SD(D_p^r)\) and \(O|SD(D_p^r)|=2r(p-1)+2\). Then, the \(SO_{\mathrm{red}}\) of \(SD(D_p^r)\) is

$ \begin{gathered} SO_{\mathrm{red}}(SD(D_p^r))=\sum_{u_i u_j \in E(SD(D_p^r))} \sqrt{(d_G(u_i)-1)^2+(d_G(u_j)-1)^2},\\ SO_{\mathrm{red}}(SD(D_p^r))=r(p^2-3)\sqrt{2(p+1)^2}+\sqrt{2(r(p+1))^2}+2r(p+1)\sqrt{(r(p+1))^2+(p+1)^2}. \end{gathered} $

Theorem 15. Consider a graph \(SD(D_p^r)\) and \(O|SD(D_p^r)|=2r(p-1)+2\). Then, the \(SO_{\mathrm{avg}}\) of \(SD(D_p^r)\) is

where, \(2m=2r(p-1)(p+2)+2(r(p+1)+1)\).

Proof. By letting \(SD(D_p^r)\) be a strong \(D(D_p^r)\) and \(O|SD(D_p^r)|=2r(p-1)+2\), where \(p\geq 3\) and \(r\geq 2\). We know,

$SO_{\mathrm{avg}}(SD(D_p^r))=\sum_{u_i u_j \in E(SD(D_p^r))} \sqrt{\left(d_G(u_i)-\frac{2m}{n}\right)^2+\left(d_G(u_j)-\frac{2m}{n}\right)^2} $

Hence, it follows the result.

By obtaining results from theorem (13), (14), and (15), we computed the relationship between the results of Sombor descriptors of a \(SD(D_p^r)\). The graphic and numeric comparison of descriptors are provided in Table 5 and Figure 11.

Table 5 confirms a consistently increasing trend for all three Sombor indices as \((p=r)\) increased from \(3\) to \(10\), with

$ SO(SD(D_p^r))>SO_{\mathrm{red}}(SD(D_p^r))>SO_{\mathrm{avg}}(SD(D_p^r)) $

throughout the considered range. From \((p,r)=(3,3)\) to \((10,10)\), \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\) increased by factors of approximately \(85.80\), \(93.68\), and \(112.69\), respectively. At \((10,10)\), the ratios calculated were \(SO_{\mathrm{red}}/SO\approx 0.961\) and \(SO_{\mathrm{avg}}/SO\approx 0.563\). The strong double construction therefore produced a further substantial increase in the magnitude of the indices. For example, at \((10,10)\), the \(SO\) and \(SO_{\mathrm{red}}\) values of the strong double graph were approximately \(65.55\) and \(80.15\) times, respectively; for those of the corresponding Dutch windmill graph, \(SO_{\mathrm{avg}}\) was approximately \(216.86\) times larger. This indicates that the additional connectivity introduced by the strong double construction has a pronounced effect on the growth of the Sombor indices.

In this study, we investigated the Sombor index (\(SO\)), reduced Sombor index (\(SO_{\mathrm{red}}\)), and average Sombor index (\(SO_{\mathrm{avg}}\)) for several special graph families, including web graphs, windmill graphs, Dutch windmill graphs, and the double and strong double graphs of Dutch windmill graphs. The key mathematical observation is that the values of these degree-based indices are strongly influenced by the degree distribution of the underlying graph. Since the Sombor type indices are defined in terms of the degrees of adjacent vertices, changes in vertex degrees and in the connectivity between vertices directly affect their numerical values.

For all the considered graph families and parameter ranges, the numerical results consistently show the ordering

$ SO(G)>SO_{\mathrm{red}}(G)>SO_{\mathrm{avg}}(G) $

Furthermore, the values of all three indices increased as the parameters of the corresponding graph families enlarged, hence reflecting the effect of the growing graph structure and its degree distribution. The double graph construction introduces additional connections between the two copies of the Dutch windmill graph and consequently produces substantially larger values of the Sombor indices. The strong double construction provides an even stronger coupling between the two copies through the adjacency relations, resulting in a further increase in the index values. Thus, the numerical comparisons demonstrate that increasing connectivity through double and strong double graph operations has a significant effect on degree-based Sombor indices.

Overall, the obtained formulas and numerical comparisons indicated that the Sombor indices provided a useful means of distinguishing the structural changes caused by different graph constructions. In particular, the changes in degree distribution and connectivity induced by the double and strong double operations are explicitly mirrored in the magnitudes and growth patterns of \(SO\), \(SO_{\mathrm{red}}\), and \(SO_{\mathrm{avg}}\).