Abstract
Sierpiński graphs are frequently related to fractals, and fractals apply in several fields of science, i.e., in chemical graph theory, computer networking, biology, and physical sciences. Functions and polynomials are powerful tools in computer mathematics for predicting the features of networks. Topological descriptors, frequently graph constraints, are absolute values that characterize the topology of a computer network. In this essay, Firstly, we compute the M-polynomials for Sierpiński-type fractals. We derive some degree-dependent topological invariants after applying algebraic operations on these M-polynomials.
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1 Introduction
A fractal represents a geometric development that’s self-comparable all through its structure. Fractals make a difference in thinking about and recognizing basic logical thoughts, such as solidifying water (snowflake) designs, microbes’ development, and brain waves. Sierpi’nski-type networks are considered in fractal hypothesis [25] and apply in various zones of science, i.e., in biology, chemical chart hypothesis, computer organizing, and physical sciences [11, 19, 20].
Functions and polynomials are powerful tools in computer mathematics for predicting the features of computer networks. Topological descriptors, frequently graph invariants, are numerical numbers that characterize the topology of a network. Multiple types of such invariants are examined and used in computer networking, pharmaceutical, and other branches of science [15, 18, 21]. The chemical structure under investigation is linked to qualities such as fracture toughness, boiling point, and heat of formation. In the discipline of chemical graph theory, this finding is significant [4]. Instead, a general method can directly produce a particular class’s different topological indices. Keeping this idea in mind, the concept of polynomials is introduced in graph theory. By constructing a general polynomial corresponding to a given structure, one can compute several topological indices by differentiating or integrating (or a different combination of both) the corresponding polynomial. Zagreb polynomials, F-polynomials, and Hosoya polynomials are all polynomials with one or two topological invariants [9, 17]. Hosoya polynomial has Wiener and super Wiener index [13].
The M-polynomial is a relatively new polynomial. It will yield innovative chemical graph discoveries and new insights into the study of degree-based topological descriptors [6]. In chemistry, algebraic polynomials [1] have a wide range of applications. The distance-related topological descriptor is calculated using the Wiener polynomial, also known as the Hosoya polynomial [12]. Similarly, when determining degree-related topological descriptors, the M-polynomial is critical [2].
2 Some literature review and basic definitions
In this paper, \({\mathcal {H}}\) is supposed to be a connected, finite and simple graph, the vertex set and edge set of graph \({\mathcal {H}}\) are \(V({\mathcal {H}})\) and \(E({\mathcal {H}})\) respectively, here \(\eta ^{v}\) denote the degree of vertex v.
Deutsch and Klavz̆ar [10] introduced the M-polynomial for \({\mathcal {H}}\) as follows
where \(\delta =\min {\eta ^{v}:v\in V({\mathcal {H}})}, \triangle =\max {\eta ^{v}:v\in V({\mathcal {H}})}\) and \(n_{cd}({\mathcal {H}})\) be the cardinality of edges \( uv\in E({\mathcal {H}})\).
The generalized Sierpiński network [7], is a graph of dimension t with vertex set \(V^t\), which is the set of all words \(v_1v_2...v_t\) of length t, where \(V=V\left( {\mathcal {H}}\right) \), denoted by \(S\left( {\mathcal {H}},t\right) \). Here \(v_p\in V\), \(1\le p \le t\), two vertices u, w associated by an edge in \(gS\left( {\mathcal {H}},t\right) \) if and only if there is \(i\in \{1,2,...,t\}\) such that
\(\bullet \) \(u_j=w_j\) if \(j<i\)
\(\bullet \) \(u_i\ne w_i\) and \({u_i,w_i}\in E\left( {\mathcal {H}}\right) \)
\(\bullet \) \(u_j=w_i\) and \(u_i=w_j\) if \(j>i\)
Above interpretation, shows that if \(uw\in E\left( S\left( {\mathcal {H}},t\right) \right) \) then \(xy\in E\left( {\mathcal {H}}\right) \) and a term z such that \(u=zxyy...y\) and \(w=zyxx...x\). A node of expression uu...u is called a extreme vertex and represented by \({\bar{w}}\). \(S\left( {\mathcal {H}},t\right) \) has always p extreme vertices, where p is the order of base graph \({\mathcal {H}}\). Furthermore, \(d_{\mathcal {H}}\left( u\right) +1=d_{S\left( {\mathcal {H}},t\right) }wuu...u\) and \(d_{\mathcal {H}}\left( w\right) +1=d_{S\left( {\mathcal {H}},t\right) }uww...w.\)
The Randic invariant [8], proposed by Milan Randić in 1975, was the first degree-related topological descriptor. It was later called molecular connectivity invariant [3] and is now known as Randic invariant. Bollobás et al. [5] refined a common form of the Randic index, stated as:
where \(\eta ^{u}\) and \(\eta ^{v}\) represents the degree of vertices u and v, and \(\alpha \in R\). Chemists and mathematicians explored this invariant broadly, the result of which, the Randic invariant has been found to have a correlation. The inverse Randic invariant \(RR_{\alpha }(X)\) is defined as:
Clearly, these invariants can be established for a variety of distinct values of \(\alpha \).
The first \(M_{1}\) and second \(M_{2}\) Zagreb invariants invented by Gutman and Trinajstić [14, 16, 26] as follows:
The modified Zagreb invariant can be defined as [27]:
The symmetric division deg index \(SDD({\mathcal {H}})\) for connected graphs defined as follows:
One more adjustment of Randić invariant is the harmonic invariant expressed as:
The inverse sum indeg \(I({\mathcal {H}})\) invariant is defined as:
In above table
\(D_{x}=\frac{\partial g(x,y)}{\partial x}\), \(D_{y}=\frac{\partial g(x,y)}{\partial y}\), \(S_{x}=\int _{0}^{x}\frac{ g(t,y)}{t}dt\), \(S_{y}=\int _{0}^{y}\frac{ g(x,t)}{t}dt\),
\(J g(x,y)= g(x,x)\), & \(Q_{\alpha } g(x,y)= x^{\alpha } g(x,y)\)
where \(D_x\) means derivative of a given function w.r.t x, while \(S_x\) represents the integral of a given function w.r.t x.
3 Main results
In this study, we constructed the M-polynomial of fractals, namely Sierpiński graphs, by considering cycle, star, and complete graph as a base graph \({\mathcal {H}}\). Moreover, we also calculate the topological indices for Sierpiński-type graphs using M-polynomial. In this article, l represents the order of a base graph \({\mathcal {H}}\).
Theorem 3.1
Let \(S(C_l,t)\) be a Sierpiński graph, whose base graph is a cycle graph. Then for \(l\ge 3\) and \(t\ge 2\)
Proof
Let \(S(C_l,t)\) be a Sierpiński graph, whose base graph is a cycle graph. Then from Fig. 1, we have
Since, type of edges of \(S(C_l,t)\) w.r.t end vertices are
Hence, the M-polynomial of \(S(C_k,t)\) is computed as
\(\square \)
Corollary 3.2
Let \(S(C_l,t)\) be a Sierpiński graph, whose base graph is a cycle graph. Then for \(l\ge 3\) and \(t\ge 2\)
Proof
Since, the M-polynomial of cycle based Sierpiński graph is
Then, for the derivation of different topological invariants, first we have to calculate the derivative, integral and combination of both operators of M-polynomial of cycle based Sierpiński network, which are as follows:
Now, we will calculate the different topological invariants by applying algebraic operations on above operators
1. First Zagreb Index:
2. Second Zagreb Index:
3. Modified Second Zagreb Index:
4. Generalized Randic Index:
5. Inverse Randic Index:
6. Symmetric Division Deg Index:
7. Harmonic Index:
8. Inverse Sum Indeg Index:
\(\square \)
Figure 2 shows that all the topological indices increased gradually with the increase of values of l and t.
Theorem 3.3
Let \(S(S_l,t)\) be a Sierpiński graph, whose base graph is a star graph. Then for \(l\ge 3\) and \(t\ge 2\)
Proof
Let \(S(S_l,t)\) be a Sierpiński graph, whose base graph is a star graph. Then from Fig. 3, we have
Since, the type of edges of \(S(S_l,t)\) w.r.t end vertices are
Hence, the M-polynomial of \(S(S_l,t)\) is
\(\square \)
Corollary 3.4
Let \(S(S_l,t)\) be a Sierpiński graph, whose base graph is a star graph. Then for \(l\ge 3\) and \(t\ge 2\)
Proof
Since, the M-polynomial of star based Sierpiński network is
Then, for the derivation of different topological invariants, first we have to calculate the derivative, integral and combination of both operators of M-polynomial of star based Sierpiński network, which are as follows:
Now, we will calculate the different topological invariants by applying algebraic operations on above operators
1. First Zagreb Index:
2. Second Zagreb Index:
3. Modified Second Zagreb Index:
4. Generalized Randic Index:
5. Inverse Randic Index:
6. Symmetric Division Deg Index:
7. Harmonic Index:
8. Inverse Sum Indeg Index:
\(\square \)
Figure 4 shows that all the topological indices increased with the increase of values of l and t.
Theorem 3.5
Let \(S(K_l,t)\) be a Sierpiński graph, whose base graph is a complete graph. Then for \(l\ge 3\) and \(t\ge 2\)
Proof
Let \(S(K_l,t)\) be a Sierpiński graph, whose base graph is a complete graph. Then from Fig. 5, we have
Since, the type of edges of \(S(K_l,t)\) w.r.t end vertices are
Hence, the M-polynomial of \(S(K_l,t)\) is computed as
\(\square \)
Corollary 3.6
Let \(S(K_l,t)\) be a Sierpiński graph, whose base graph is a complete graph. Then for \(l\ge 3\) and \(t\ge 2\)
Proof
Since, the M-polynomial of complete based Sierpiński network is
Then, for the derivation of different topological invariants, first we have to calculate the derivative, integral and combination of both operators of M-polynomial of cycle based Sierpiński network, which are as follows:
Now, we will calculate the different topological invariants for complete based Sierpiński graph by applying algebraic operations on above operators
1. First Zagreb Index:
2. Second Zagreb Index:
3. Modified Second Zagreb Index:
4. Generalized Randic Index:
5. Inverse Randic Index:
6. Symmetric Division Deg Index:
7. Harmonic Index:
8. Inverse Sum Indeg Index:
\(\square \)
Figure 6 shows that all the topological indices increased with the increase of values of l and t.
4 Conclusion
In this study, firstly, we obtained the M-polynomials of Sierpiński-type networks using cycle, star, and complete network as a base structure. Topological descriptors make it simpler and more accurate to identify these networks’ physical characteristics, chemical reactivity, and biological activities. Therefore, we calculate the formulae for some degree-related topological invariants, such as the first, second, modified, general, and inverse Randić invariants, symmetric division invariant, harmonic invariants and inverse sum invariant for Sierpiński graphs by using M-polynomial. Moreover, we compare the different topological invariants using a graph.
Future work
The “modified symmetric division deg index" and “harmonic-arithmetic index" are newly introduced invariants presented in [22,23,24]. In our next study, we will obtain these invariants for Sierpiński-type networks and compare them with the “symmetric division deg index" and “harmonic index."
Data Availability
Not applicable.
References
Ajmal, M.; Nazeer, W.; Munir, M.; Kang, S.M.; Jung, C.Y.: The M-polynomials and topological indices of generalized prism network. Int. J. Math. Anal. 11(6), 293–303 (2017)
Ali, A.; Nazeer, W.; Munir, M.; Kang, S.M.: M-polynomials and topological indices of zigzag and rhombic benzenoid systems. Open Chem. 16(1), 73–78 (2018)
Amić, D.; Bešlo, D.; Lucić, B.; Nikolić, S.; Trinajstić, N.: The vertex-connectivity index revisited. J. Chem. Inf. Comput. Sci. 38(5), 819–822 (1998)
Baig, A.Q.; Imran, M.; Ali, H.: On topological indices of poly oxide, poly silicate, DOX, and DSL networks. Can. J. Chem. 93(7), 730–739 (2015)
Bollobás, B.; Erdos, P.: Graphs of extremal weights. Ars Combin. 50, 225–233 (1998)
Chu, Y.M.; Imran, M.; Baig, A.Q.; Akhter, S.; Siddiqui, M.K.: On M-polynomial based topological descriptors of chemical crystal structures and their applications. The European Physical Journal Plus 135(11), 1–19 (2020)
Cristea, L.I.; Steinski, B.: Distances in Sierpiński graphs and on the Sierpiński gasket. Aequat. Math. 85, 201–219 (2013)
Deng, H.; Yang, J.; Xia, F.: A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes. Comput. Math. Appl. 61(10), 3017–3023 (2011)
Das, K.C.; Gutman, I.: Some properties of the second Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry 52(1), 3–1 (2004)
Deutsch, E., Klavžar, S.: M-polynomial and degree-based topological indices. arXiv preprint arXiv:1407.1592 (2014)
Estrada-Moreno, A.; Rodríguez-Velázquez, J.A.: On the General Randić index of polymeric networks modelled by generalized Sierpiński graphs. Discret. Appl. Math. 263, 140–151 (2019)
Eliasi, M.; Taeri, B.: Hosoya polynomial of zigzag polyhex nanotorus. J. Serb. Chem. Soc. 73(3), 311–319 (2008)
Farahani, M.R.: Hosoya polynomial, Wiener and Hyper Wiener indices of some regular graphs. Informatics Engineering, an International Journal 1(1), 9–13 (2013)
Gutman, I.; Das, K.C.: The first Zagreb index 30 years after. MATCH Communications in Mathematical and in Computer Chemistry 50(1), 83–92 (2004)
Gutman, I.; Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, New York (2012)
Gutman, I.; Trinajstić, N.: Graph theory and molecular orbitals. Total phi-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17(4), 535–538 (1972)
Hosoya, H.: On some counting polynomials in chemistry. Discret. Appl. Math. 19(1–3), 239–257 (1988)
Hayat, S.; Malik, M.A.; Imran, M.: Computing topological indices of honeycomb derived networks. Romanian Journal of Information Science and Technology 18, 144–165 (2015)
Imran, M.; Gao, W.; Farahani, M.R.: On topological properties of sierpiński networks. Choas Soliton & Fractals 98, 199–204 (2017)
Klavžar, S.; Milutinović, U.; Petr, C.: 1-perfect codes in Sierpiński graphs. Bull. Aust. Math. Soc. 66(3), 369–384 (2002)
Liu, J.B.; Pan, X.F.; Hu, F.T.; Hu, F.F.: Asymptotic Laplacian-energy-like invariant of lattices. Appl. Math. Comput. 253, 205–214 (2015)
Liu, J. B., Bao, Y., Zheng, W. T.: Analyses of some structural properties on a class of hierarchical scale-free networks. Fractals, 30(07), (2022)
Liu, J. B., Bao, Y., Zheng, W. T., Hayat, S.: Network coherence analysis on a family of nested weighted n-polygon networks. Fractals, 29(08), (2021)
Liu, J. B., Zhao, J., Min, J., Cao, J.: The Hosoya index of graphs formed by a fractal graph. Fractals, 27(08), (2019)
Teplyaev, A.: Spectral analysis on infinite Sierpiński gaskets. J. Funct. Anal. 159(2), 537–567 (1998)
Trinajstić, N.; Nikolić, S.; Miličević, A.; Gutman, I.: About the zagreb indices. Kemija u Industriji: Časopis kemičara i kemijskih inženjera Hrvatske 59(12), 577–589 (2010)
Vukičević, D.; Graovac, A.: Valence connectivity versus Randić, Zagreb and modified Zagreb index: A linear algorithm to check discriminative properties of indices in acyclic molecular graphs. Croat. Chem. Acta 77(3), 501–508 (2004)
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