Degree sequence of the generalized Sierpinski graph

TL;DR

This paper calculates the degree sequence and the first Zagreb index of the generalized Sierpinski graph based on the parameters of its base graph G, providing a mathematical characterization of its structure.

Contribution

It introduces formulas for the degree sequence and Zagreb index of the generalized Sierpinski graph in relation to the base graph G, advancing understanding of its properties.

Findings

Degree sequence expressed in terms of base graph parameters

First Zagreb index derived for the generalized Sierpinski graph

Provides a mathematical framework for analyzing these graphs

Abstract

We determine the degree sequence of the generalized Sierpinski graph and its general first Zagreb index in terms of the same parameters of the base graph G.

Figures

Equations

$$\frac{\epsilon (T)(n^t-2n^{t-1}+1)}{n-1}$$

$$\displaystyle\deg_{S(G,t)}(x_{1}x_{2}...x_{t})=\left\{\begin{array}[]{ll}1+\deg_{G}(x_{t})&~{}~{}x_{1}x_{2}...x_{t}\neq x_{t}x_{t}...x_{t}~{}and~{}~{}x_{i}\in N_{G}(x_{t})~{}for\\ &~{}~{}i={\max\{j:~{}1\leq j\leq t-1,~{}x_{j}\neq x_{t}\}},\\ \\ \deg_{G}(x_{t})&~{}~{}otherwise.\end{array}\right.$$

$$\mathbf{x}=x_1{x}_2...x_{i-1}{x}_i{z}_i{z}_i...z_i,\mathbf{z}=x_1{x}_2...x_{i-1}{z}_i{x}_i{x}_i...x_i$$

$$\mathbf{x}=x_1{x}_2...x_{i-1}{x}_i{x}_t{x}_t...x_t,\mathbf{z}=x_1{x}_2...x_{i-1}{x}_t{x}_i{x}_i...x_i$$

$$N_{S(G,t)}(x_{1}x_{2}...x_{t})\!=\!\left\{\begin{array}[]{ll}\!\!\!\{{\bf y_{1}},{\bf y_{2}},...,{\bf y_{d}},x_{1}x_{2}...x_{i-1}x_{t}x_{i}x_{i}...x_{i}\}&~{}~{}x_{1}x_{2}...x_{t}\neq x_{t}x_{t}...x_{t},~{}x_{i}\in N_{G}(x_{t}),\\ \!\!\!\{{\bf y_{1}},{\bf y_{2}},...,{\bf y_{d}}\}&~{}~{}otherwise,\end{array}\right.$$

$$|V_{k}|~{}n^{t-1}-{n^{t-1}-1\over n-1}~{}\big{(}k~{}|V_{k}|-(k-1)~{}|V_{k-1}|\big{)}.$$

$${\Omega }_{s,x}=\{v_1{v}_2...v_t\in V(S(G,t)):v_t=x\}$$

$${\Gamma }_{s,x,i}=\{x_1{x}_2...x_t\in V(S(G,t)):x_i\notin N_G(x)\cup \{x\},x_j=x\forall j>i\}.$$

$$v_1{v}_2...v_{t-1}x\in \left({\cup }_{i=1}^{t-1}{\Gamma }_{s,x,i}\right)\cup \{xx...x\}.$$

$$\big{|}\left(\cup_{i=1}^{t-1}~{}\Gamma_{s,x,i}\right)\cup\{xx...x\}\big{|}=1+(n-s-1)~{}(1+n+n^{2}+\cdots+n^{t-2}),$$

$$|V_{s}|~{}\big{(}1+(n-s-1)~{}(1+n+n^{2}+\cdots+n^{t-2})\big{)}=|V_{s}|~{}\big{(}n^{t-1}-s~{}(1+n+\cdots+n^{t-2})\big{)}.$$

$$\displaystyle|\Omega_{s}|-|V_{s}|~{}\big{(}n^{t-1}-s~{}(1+n+\cdots+n^{t-2})\big{)}$$

$$|V_{k}|~{}\big{(}n^{t-1}-k~{}(1+n+\cdots+n^{t-2})\big{)}+(k-1)~{}|V_{k-1}|~{}(1+n+\cdots+n^{t-2})$$

$$Z_{\alpha }(G)=\sum _{\{u,v\}\in E(G)}\left(({\mathrm{deg}}_G(u){)}^{\alpha -1}+({\mathrm{deg}}_G(v){)}^{\alpha -1}\right)=\sum _{u\in V(G)}({\mathrm{deg}}_G(u){)}^{\alpha }$$

$$Z_{\alpha }(S(G,t))=\frac{n^t-n^{t-1}+(n^{t-1}-1)\alpha }{n-1}{Z}_{\alpha }(G)+\frac{n^{t-1}-1}{n-1}\sum _{j=1}^{\alpha -1}\left(\genfrac{}{}{0ex}{}{\alpha }{j-1}\right)Z_j(G).$$

$$Z_{\alpha }(S(G,t))$$

$$|V(S(G,t))|=Z_0(S(G,t))=n^t,|E(S(G,t))|=\frac{1}{2}{Z}_1(S(G,t))=\frac{n^t-1}{n-1}|E(G)|$$

$$M_1(S(G,t))=Z_2(S(G,t))=\frac{n^t+n^{t-1}-2}{n-1}{Z}_2(G)+\frac{n^{t-1}-1}{n-1}2|E(G)|$$

$$F(S(G,t))=Z_3(S(G,t))=\frac{n^t+2n^{t-1}-3}{n-1}F(G)+\frac{n^{t-1}-1}{n-1}(2|E(G)|+3M_1(G)).$$

Full text

Key words: Degree sequence, Generalized Sierpiński graph, General first Zagreb index.

2010 Mathematics Subject Classification: 05C07, 05C76.

1 Introduction

All graphs considered in this paper are assumed to be simple and finite. Throughout the paper, $G=(V,E)$ will denote a graph of order $n=|V|$ with the vertex set $V$ and the edge set $E$. The degree of a vertex $v$ of $G$ is denoted by $\deg_{G}(v)$ which is the size of the set of its neighbourhood $N_{G}(v)$. Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures, especially when the considered structures have self-similarity properties. In these cases we typically only need to study the substructures and the way that they are related together. For example polymer networks can be modeled by generalized Sierpiński graphs, see [9]. Sierpiński and Sierpiński-type Graphs are studied in fractal theory [38] and appear naturally in diverse areas of mathematics and in several scientific fields. This family of graphs were studied for the first time in [23] and [32], independently, and constitutes an extensively studied class of graphs of fractal nature with applications in computer science (as a model for interconnection networks which are known as WK-recursive networks), topology and mathematics of the Tower of Hanoi, see [6] and [36] for more details. One of the most important families of such graphs is formed by the Sierpiński gasket graphs introduced by Scorer, Grundy and Smith in [37] which play an important role in psychology, dynamic systems and probability, see [14], [22] and [28]. Sierpiński, Sierpiński-type and generalized Sierpiński graphs have many interesting properties and were studied extensively in literature.

Definition 1.1

[13*]** Let $G=(V,E)$ be a graph of order $n\geq 2$, $t$ be a positive integer and denote the set of words of length $t$ on the alphabet $V$ by $V^{t}$. The letters of a word ${\bf u}\in V^{t}$ (of length $t$) are denoted by $u_{1}u_{2}...u_{t}$. The generalized Sierpiński graph of $G$ of dimension $t$, denoted by $S(G,t)$, is the graph with vertex set $V^{t}$ and $\{{\bf u},{\bf v}\}$ is an edge in it if and only if there exists $i\in\{1,...,t\}$ such that:

(i)  $u_{j}=v_{j}~{}~{}~{}~{}if~{}~{}~{}~{}j<i$,

(ii)  $u_{i}\neq v_{i}~{}~{}~{}~{}~{}and~{}~{}~{}\{u_{i},v_{i}\}\in E(G)$,

(iii)  $u_{j}=v_{i}~{}~{}~{}and~{}~{}~{}v_{j}=u_{i}~{}~{}if~{}~{}j>i$.*

For example $S(C_{4},3)$ is depicted in Figure 1 in which $C_{4}$ is assumed to be a cycle with the vertex set $\{1,2,3,4\}$ and the edge set $\{\{1,2\},\{2,3\},\{3,4\},\{4,1\}\}$. Note that $S(G,1)$ is (isomorphic to) the base graph $G$ and $S(G,2)$ can be constructed by copying $n$ times $S(G,1)$ and adding an edge between the $i$-th vertex of the $j$-th copy and the $j$-th vertex of the $i$-th copy of $S(G,1)$ whenever $\{i,j\}$ is an edge in $G$. In fact $S(G,t)$ is a fractal-like graph that uses $G$ as a building block. When $G$ is the complete graph $K_{n}$, the (ordinary) Sierpiński graph $S^{t}_{n}=S(K_{n},t)$ is obtained. Klav$\check{z}$ar et al. introduced the graph $S(K_{n},t)$ for the first time and they show that $S(K_{3},t)$ is isomorphic to the graph of the Tower of Hanoi $H^{t}_{3}$, see [23] and [24]. It is well known that $S^{t}_{n}$ contains $n$ (extreme) vertices of degree $n-1$ and all the other vertices are of degree $n$, see [16]. Hence, $S^{t}_{n}$ is almost regular. The degree sequence of the Hanoi graph $H^{t}_{n}$ (the state graph of the Tower of Hanoi game with $n$ pegs and $t$ discs) is more complex and is completely determined in [18]. In this paper, we want to determine the degree sequence of the generalized Sierpiński graph $S(G,t)$ of an arbitrary graph $G$. Sierpiński graphs are studied from numerous points of view. In [23] and [41] shortest paths in Sierpiński graphs are studied. In [15] an algorithm is proposed which makes use of three automata to determine all shortest paths in Sierpiński graphs. Metric properties of Sierpiński graphs is investigated in [20] and [31]. For connections between the Sierpiński graphs and Stern’s diatomic sequence see [17]. Identifying codes, locating-dominating codes and total-dominating codes in Sierpiński graphs are studied in [12]. Sierpiński graphs contain (essentially) unique 1-perfect codes [24]. Covering codes in Sierpiński graphs is studied in [3] and equitable $L(2,1)$-labelings of them is considered in [10]. In [26] the canonical isometric representation of Sierpiński graphs is explicitly described. The crossing number of Sierpiński graphs is studied in [25], giving the first infinite families of graphs of fractal nature for which the crossing number is determined (up to the crossing number of complete graphs). Colorings and the chromatic number of these graphs is studied in [19] and the hub number of them is determined in [29].

Also, many papers studied the structure of the generalized Sierpiński graphs. In [13] some interesting results about the generalized Sierpiński graphs (concerning their automorphism groups, perfect codes and distinguishing numbers) are obtained. The total chromatic number for some families of these graphs is determined by Geetha and Somasundaram in [11]. More precisely, the authors prove the tight bound of the Behzad and Vizing conjecture on total coloring for the generalized Sierpiński graphs of cycle graphs and hypercube graphs. They provide a total coloring for the WK-recursive topology, which also gives the tight bound. In [8] the distance between vertices of $S(G,t)$ is expressed in terms of the distance between vertices of the base graph $G$. In addition, the authors give an explicit formula for the diameter and the radius of $S(G,t)$ when the base graph $G$ is a tree. In [34] their independence number, chromatic number, vertex cover number, clique number and domination number are investigated in terms of the similar parameters of the base graph $G$. The strong metric dimension of these graphs is studied in [7]. Metric properties of generalized Sierpiński graphs over stars are considered in [2]. The Roman domination number of $S(G,t)$ is investigated in [33]. An explicit formula for the number of connected components of $S(G,t)$ is given in [27] and it is proved that the (edge-)connectivity of $S(G,t)$ equals the (edge-) connectivity of $G$. Also, It is demonstrated that $S(G,t)$ contains a perfect matching if and only if $G$ contains a perfect matching. Moreover, Hamiltonicity of these graphs is also discussed there. It is shown in [34] that for any tree $T$ of order $n\geq 2$ and any positive integer $t$, $S(T,t)$ is a tree and the number of leaves of $S(T,t)$ is equal to

[TABLE]

in which $\varepsilon(T)$ is the number of leaves of $T$. We generalize this result in Theorem 2.1. For more results in these subject and related subjects, see [13], [16], [18], [21], [34] and [35].

2 Main results

First, we determine the neighbourhood of a vertex in $S(G,t)$ and hence, its degree in $S(G,t)$. Then, for each $0\leq k\leq\Delta(S(G,t))$, we determine the number of vertices of degree $k$ in $S(G,t)$ which leads to the degree sequence of the generalized Sierpiński graph $S(G,t)$. Finally, we show that the general first Zagreb index of $S(G,t)$ can be expressed as a linear combination of the general first Zagreb indices of the base graph $G$.

Lemma 2.1

Let $G=(V,E)$ be a simple graph and $t\geq 1$ be an integer. Then, for each vertex ${\bf x}=x_{1}x_{2}...x_{t}$ in the generalized Sierpiński graph $S(G,t)$ we have

[TABLE]

**Proof. **

Obviously the result follows for $t=1$ because $x_{1}...x_{t}=x_{t}...x_{t}=x_{t}$ and $\deg_{S(G,1)}(x_{1})=\deg_{G}(x_{1})$. Hence, we assume that $t\geq 2$. By the adjacency rule in $S(G,t)$, it is straightforward to see that each vertex ${\bf x^{\prime}}=x^{\prime}_{1}x^{\prime}_{2}...x^{\prime}_{t}$ in $S(G,t)$ with $N_{G}(x^{\prime}_{t})=\emptyset$ is an isolated vertex in $S(G,t)$. Thus, If $\deg_{G}(x_{t})=0$, then ${\bf x}=x_{1}x_{2}...x_{t}$ is an isolated vertex in $S(G,t)$ and the result directly follows. Assume that $\deg_{G}(x_{t})=d\geq 1$ and $N_{G}(x_{t})=\{y_{1},y_{2},...,y_{d}\}$. For each $i\in\{1,2,...,d\}$, it is easy to check that two vertices ${\bf x}=x_{1}x_{2}...x_{t-1}x_{t}$ and ${\bf y_{i}}=x_{1}x_{2}...x_{t-1}y_{i}$ are adjacent in $S(G,t)$. Hence, $\deg_{S(G,t)}({\bf x})\geq d=\deg_{G}(x_{t})$. Now let ${\bf z}=z_{1}z_{2}...z_{t}$ be a neighbour of ${\bf x}$ in $S(G,t)$. Thus, by the adjacency rule in $S(G,t)$, there exists $i\in\{1,2,...,t\}$ such that $x_{j}=z_{j}$ for each $j<i$, $\{x_{i},z_{i}\}\in E(G)$ and ($x_{\ell}=z_{i}$ and $z_{\ell}=x_{i}$) for each $\ell>i$. If $i=t$, then ${\bf z}=x_{1}x_{2}...x_{t-1}z_{t}$ and $\{x_{t},z_{t}\}\in E(G)$. This implies that $z_{t}\in N_{G}(x_{t})=\{y_{1},y_{2},...,y_{d}\}$ and hence ${\bf z}\in\{{\bf y_{1}},{\bf y_{2}},...,{\bf y_{d}}\}$. If $i<t$, then we must have

[TABLE]

Thus, $x_{t}=z_{i}$ and $x_{i}\in N_{G}(z_{i})=N_{G}(x_{t})=\{y_{1},y_{2},...,y_{d}\}$. This implies that

[TABLE]

Note that $i=\max\{j:~{}1\leq j\leq t-1,~{}x_{j}\neq x_{t}\}$. Since the vertex ${\bf x}$ is given and its structure is specified, the index $i$ is unique. Therefore, we have

[TABLE]

in which $i={\max\{j:~{}1\leq j\leq t-1,~{}x_{j}\neq x_{t}\}}$. This completes the proof.

•  *

Let $G$ be a graph of order $n\geq 2$ with $E(G)\neq\emptyset$ such that $\delta(G)=\deg_{G}(x)$, $\Delta(G)=\deg_{G}(y)$ and $z\in N_{G}(y)$. Lemma 2.1 implies that $\deg_{S(G,t)}(xx...x)=\deg_{G}(x)=\delta(G)$ and for $t\geq 2$ we have $\deg_{S(G,t)}(zz...zy)=1+\deg_{G}(y)=1+\Delta(G)$. Hence, the following corollary directly follows from Lemma 2.1, (also, see [34]).

Corollary 2.2

Let $G$ be a simple graph of order $n\geq 2$ and $t\geq 1$ be an integer. Then,

• i)

$\delta(S(G,t))=\delta(G)$.

• ii)

$\Delta(S(\overline{K_{n}},t))=\Delta(\overline{K_{n}})=0$* and $\Delta(S(G,1))=\Delta(G)$. Also, we have $\Delta(S(G,t))=1+\Delta(G)$ when $E(G)\neq\emptyset$ and $t\geq 2$,*

Theorem 2.1

Let $G=(V,E)$ be a simple graph of order $n\geq 2$, $t\geq 1$ be an integer and for each $k$, $\delta(G)\leq k\leq\Delta(G)$, let $V_{k}=\{v\in V:~{}\deg_{G}(v)=k\}$. Then, for each $k$, $\delta(S(G,t))\leq k\leq\Delta(S(G,t))$, the number of vertices of degree $k$ in the generalized Sierpiński graph $S(G,t)$ is

[TABLE]

**Proof. **

It is easy to check that the result directly follows for the case $t=1$. Hereafter we assume that $t\geq 2$. For each integer $s$, $\delta(G)\leq s\leq\Delta(G)$, and for each $x\in V_{s}$ define

[TABLE]

and let $\Omega_{s}=\cup_{x\in V_{s}}\Omega_{s,x}$. Note that $|\Omega_{s,x}|=n^{t-1}$ and $|\Omega_{s}|=|V_{s}|~{}n^{t-1}$. By Lemma 2.1, the degree of each vertex in $\Omega_{s}$ is $s$ or $s+1$. Now we want to determine the number of vertices in $\Omega_{s}$ which has degree $s$. Let $x\in V_{s}$ and $v_{1}v_{2}...v_{t-1}x\in\Omega_{s}$. If $\deg_{S(G,t)}(v_{1}v_{2}...v_{t-1}x)=s$, then by Lemma 2.1 we have $v_{1}v_{2}...v_{t-1}x=xx...x$ or there exists $1\leq i\leq t-1$ such that $i={\max\{j:~{}1\leq j\leq t-1,~{}v_{j}\neq x\}}$ and $v_{i}\notin N_{G}(x)$. For each $i$, $1\leq i\leq t-1$, define

[TABLE]

Note that $|\Gamma_{s,x,i}|=n^{i-1}~{}(n-s-1)$. Thus, the degree of the vertex $v_{1}v_{2}...v_{t-1}x\in\Omega_{s}$ is equal to $s$ if and only if

[TABLE]

Since $x\in V_{s}$ and

[TABLE]

the number of vertices in $\Omega_{s}$ of degree $s$ is

[TABLE]

Hence, the number of vertices in $\Omega_{s}$ of degree $s+1$ is given by

[TABLE]

Therefore, the number of vertices of degree $k$ in the generalized Sierpiński graph $S(G,t)$ is equal to

[TABLE]

Since $V_{{}_{1+\Delta(G)}}=\emptyset=V_{{}_{-1}}$ and $1+n+\cdots+n^{t-2}={n^{t-1}-1\over n-1}$, the proof is completed.

•  *

A large number of properties like chemical activity, biological activity, physicochemical properties, thermodynamic properties are determined by the chemical applications of graph theory. These properties can be expressed by certain graph invariants (real numbers related to a graph which is structurally invariant) referred to as topological indices. Some of topological indices of Sierpiński networks and generalized Sierpiński graphs are determined, see [9], [21] and [35]. In [39] and [40] Li et al. considered the general first Zagreb index of a graph $G$ as

[TABLE]

in which $\alpha$ is a real number. Specially, we see that $Z_{0}(G)=n$, $Z_{1}(G)=\sum_{k=1}^{\Delta(G)}~{}|V_{k}|~{}k=2|E(G)|$, $Z_{2}(G)=M_{1}(G)$ which is known as the first Zagreb index and $Z_{3}(G)=F(G)$ which is known as the forgotten topological index, see [1], [4] and [30] for more details.

Corollary 2.3

For each simple graph $G$ of order $n\geq 2$ and each integer $\alpha\geq 0$, the general first Zagreb index of the generalized Sierpiński graph $S(G,t)$, $t\geq 1$, is given by

[TABLE]

**Proof. **

By the definition of the general first Zagreb index, binomial expansion formula and Theorem 2.1, we have

[TABLE]

Note that by using the generalized form of the binomial theorem, Corollary 2.3 can be motivated in such a way that $\alpha$ be a real number but then the finite sum should be replaced by an infinite series. By using Corollary 2.3, it can be easily seen that

[TABLE]

which coincides with the previously obtained results, see [34]. Also, we have

[TABLE]

and

[TABLE]

In [5], by using the Stirling numbers of the first kind, it is shown that for each integer $\alpha\geq\Delta(G)$, the general first Zagreb index $Z_{\alpha}(G)$ can be expressed as a linear combination of $Z_{0}(G)$, $Z_{1}(G)$, …, $Z_{\Delta(G)-1}(G)$. This result using Corollary 2.3 implies that $Z_{\alpha}(S(G,t))$ can also be expressed as a linear combination of $Z_{0}(G)$, $Z_{1}(G)$, …, $Z_{\Delta(G)-1}(G)$ for each $\alpha\geq\Delta(G)$.

The authors declare that they have no competing interests.