A note on the Independent domination polynomial of zero divisor graph of rings

All graphs considered in this note are finite, simple and undirected graphs. A graph is usually symbolized by $\displaystyle G=G(V(G),E(G))$ with its vertex set $\displaystyle V(G)$ and edge set $\displaystyle E(G)$. The numbers $\displaystyle n=|V(G)|$ is order and $\displaystyle m=|E(G)|$ size of $\displaystyle G$ An edge among vertices $\displaystyle u$ and $\displaystyle v$ is denoted by $\displaystyle u\sim v$. The degree $\displaystyle d_{v_{i}}(G)$ of a vertex $\displaystyle v_{i}\in V(G)$ is the number of vertices incident on it. The union of two graphs $\displaystyle G_{1}(V_{1}(G_{1}),E_{1}(G_{1}))$ and $\displaystyle G_{2}(V_{2}(G_{2}),E_{2}(G_{2}))$, denoted by $\displaystyle G_{1}\cup G_{2}$, defined as a graph with vertex set $\displaystyle V_{1}(G_{1})\cup V_{2}(G_{2})$ and edge set $\displaystyle E_{1}\cup E_{2}.$ The join of $\displaystyle G_{1}(V_{1}(G_{1}),E_{1}(G_{1}))$ and $\displaystyle G_{2}(V_{2}(G_{2}),E_{2}(G_{2}))$, denoted by $\displaystyle G_{1}\vee G_{2}$, is a graph with vertex set $\displaystyle V_{1}(G_{1})\cup V_{2}(G_{2})$ and edge set $\displaystyle E(G_{1})\cup E(G_{2})\cup\big{\{}u,v~|~u\in V(G_{1}),v\in V(G_{2})\big{\}}.$

A set $\displaystyle\emptyset\neq S\subseteq V(G)$ is called a dominating set if each vertex in $\displaystyle V\setminus S$ is adjacent to at least one vertex in $\displaystyle S.$ The minimum order (cardinality) among all such dominating sets of $\displaystyle G$ is called the domination number $\displaystyle\gamma(G)$ of $\displaystyle G.$ The theory of domination of graphs is well studied, see textbook [14]. A set $\displaystyle S\subseteq V(G)$ in a graph $\displaystyle G$ is called an independent set if vertices of $\displaystyle S$ are pairwise non-adjacent. The cardinality of the maximum independent set is called the independence number $\displaystyle\alpha(G)$ of $\displaystyle G.$ A subset $\displaystyle D$ of $\displaystyle V(G)$ is independent dominating set of $\displaystyle G$ which is both dominating and independent in $\displaystyle G$. The independent domination number $\displaystyle\gamma_{i}(G)$ is the minimum size of all independent dominating sets of $\displaystyle G.$ The invariants $\displaystyle\gamma,\alpha$ and $\displaystyle\gamma_{i}$ of $\displaystyle G$ is related by $\displaystyle\gamma(G)\leq\gamma_{i}(G)\leq\alpha(G)$ (see, [14]). The independent set problem in a graph is strongly NP-hard problem while the dominating set problem of a graph is NP-complete problem. These problems are well studied both in theoretical computer science and mathematics.

A polynomial $\displaystyle p(x)=\sum_{i=0}^{b}a_{i}x^{i}$ is called unimodal if coefficients $\displaystyle a_{i}$’s form a unimodal sequence, that is, there exists a positive integer $\displaystyle p~(0\leq p\leq n),$ known as the mode, such that $\displaystyle a_{0}\leq a_{1}\leq\dots\leq a_{p}\geq a_{p+1}\geq\dots\geq a_{b}$. Equivalently the coefficients of $\displaystyle p(x)$ increase to some stage and then decrease from thereafter. The polynomial $\displaystyle p(x)$ is log-concave (logarithmically concave) if

If $\displaystyle a_{i}$’s are non negative and all the zeros of $\displaystyle p(x)$ are real. Then the basic approach for unimodal and log-concave is the Newton’s inequalities [15] stated as:

The condition for log-concave given in (1.2) is stronger than the one given in (1.1), see [21]. Also log-concave sequence of positive terms is unimodal [21]. However, if $\displaystyle a_{i}=0$ implies that either $\displaystyle a_{1}=\dots=a_{i-1}=0$ or $\displaystyle a_{i+1}=\dots=a_{n}=0,$ (no internal zeros), then a non-negative log-concave sequence is unimodal [21]. Log-concavity of sequences is related to the surface embedding of graphs in topological graph theory, like the log-concavity of genus polynomials of graphs [12].

Clearly, the polynomial $\displaystyle 1+2x+3x^{2}+7x^{3}+6x^{4}+3x^{5}$ is unimodal whereas $\displaystyle 21+7x+19x^{2}+6x^{3}+4x^{4}$ is not unimodal, since the coefficients decrease, then increase and then decrease. An effective information about the increase or decrease of coefficients can measure the unimodal property. As it is clear that unimodal sequence either increases or decreases, or increase and then decrease. The number of changes of directions (increasing or decreasing) of $\displaystyle p(x)$ is defined as the of oscillations $\displaystyle\eta(p(x))$ of $\displaystyle p(x)$. Obviously, for the unimodal polynomial the oscillations of $\displaystyle p(x)$ must be at most one. For polynomial $\displaystyle p(x)=8+6x+160x^{2}+28x^{3}+16x^{4}+11x^{5}$, then $\displaystyle\eta(p(x))=2$, since change of decreasing directions are two. Sometimes it is equally good to identify the oscillations for non unimodal polynomials.

Identifying unimodal or log-concave property (or both) of a given polynomial is a very well studied non-trivial problem. Several open problems and conjectures exist in literature related to these entities for different types of polynomials. These properties of polynomials are mostly studied for graph polynomial, like independent polynomial, dominating polynomial, matching polynomial, clique polynomial and several other polynomials.

Let $\displaystyle d_{k}(G,k)$ be the number of independent dominating sets of order $\displaystyle k$ in $\displaystyle G$. The independent domination polynomial of $\displaystyle G$ is defined as

A root of the equation $\displaystyle D_{i}(G,x)=0$ is called as the independent domination root of $\displaystyle G.$

The independent domination polynomial $\displaystyle D_{i}(G,x)$ is a generating function of number of the independent dominating sets of certain cardinalities of $\displaystyle G$. The independent domination polynomials and their roots, unimodal and log-concave property have attracted many researchers, see [5, 19, 10, 11]. Jahari and Alikhani [16] obtained the independent domination polynomials of generalized compound graphs and constructed graphs whose independent domination polynomials have real zeros. More about independent domination polynomials and other polynomials can be seen in [2, 1, 3, 17, 18, 4]. Recently, Gürsoy, Ülker and Gürsoy [13] presented the results related to the independent domination polynomial of zero-divisors graphs associated to commutative rings. In particular, they showed that the independent domination polynomial of zero divisor graph of $\displaystyle\mathbb{Z}_{n}$ is unimodal and log-concave. We will consider this study in the present note and modify the existing results of Gürsoy, Ülker and Gürsoy [13] and fix the errors in the published article.

In Section 2, we give basic of zero divisor graphs of $\displaystyle\mathbb{Z}_{n}$ and recall the results of Gürsoy, Ülker and Gürsoy [13] along with the properties of log-concave and unimodal. We give several examples having complex independent domination zeros countering a result of [13]. We discuss the unimodal and log-concave property of $\displaystyle\Gamma(\mathbb{Z}_{n})$ for $\displaystyle n\in\{p^{2},2p,pq,p^{2}q,pqr\}$ where $\displaystyle p,q,r$ are primes in Propositions 2.2, 2.3 and 2.4. Theorems 2.6 gives the independent domination polynomial in simplified form and Theorem 2.7 established the non unimodal and log-concave property of $\displaystyle\Gamma(\mathbb{Z}_{p^{\alpha}})$ where $\displaystyle p>2$ is prime and $\displaystyle\alpha$ is a positive integer.

For a given commutative ring $\displaystyle R$ with non-zero identity, $\displaystyle 0\neq a\in R$ is a zero divisor of $\displaystyle R$ if there is $\displaystyle 0\neq b\in R$ such that $\displaystyle a\cdot b=0.$ Beck in 1988 [8] put forward the concept of zero divisor graphs $\displaystyle\Gamma(R)$ to study the colouring of rings. He included identity $\displaystyle 0$ in the vertex set of zero divisor graphs and defined edges among zero divisor if and if their product is zero, thereby we see that $\displaystyle 0$ is adjacent to all vertices of such a graphs. Anderson and Livingston in 1999 [7] modified the definition Beck’s zero divisor graphs and excluding $\displaystyle 0$ of ring in zero divisor graph and defined edges between two non-zero zero divisors if and only if their product is zero. Several interesting results related to these graphs and their underlying rings were published in the past years, mostly recent notably works are:  Chattopadhyay Panigrahi [9] obtained the graphical structure of $\displaystyle\Gamma(\mathbb{Z}_{n})$ and showed that it is join of certain complete and null graphs (non empty graphs without any edges). Anderson and Weber [6] determine all zero-divisor graphs with at most $\displaystyle 14$ vertices. First spectral analysis of zero divisor graphs were carried in [22]. Complete spectral and structural analysis were given in [20]. Gürsoy, Ülker and Gürsoy [13] introduced the concept of independent domination polynomials for zero divisor graphs for $\displaystyle\mathbb{Z}_{n}$ for certain values of $\displaystyle n.$ We will carry the work Gürsoy, Ülker and Gürsoy [13] forward in this note. First we understand the structure of $\displaystyle\Gamma(\mathbb{Z}_{n})$.

Consider the sets as given below:

where $\displaystyle t\notin\{1,n\}$ is a divisor of $\displaystyle n$ and $\displaystyle(x,n)$ denotes the greatest common divisor of $\displaystyle x$ and $\displaystyle n.$ It is clear that $\displaystyle A_{e_{i}}\cap A_{e_{j}}=\emptyset$, for $\displaystyle i\neq j$. Thus, it implies that $\displaystyle A_{e_{1}},A_{e_{2}},\dots,A_{e_{t}}$ are pairwise disjoint and partitions the vertex set of $\displaystyle\Gamma(\mathbb{Z}_{n})$ as

and vertices of $\displaystyle A_{e_{i}}$ are adjacent to vertices of $\displaystyle A_{e_{j}}$ in if and only if $\displaystyle e_{i}e_{j}=kn,$ where $\displaystyle k$ is scaler (see [9]). Also, $\displaystyle|A_{e_{i}}|=\phi\left(\frac{n}{e_{i}}\right)$, for $\displaystyle 1\leq i\leq t$ (see [22]). Besides the induced subgraphs of $\displaystyle A_{e_{i}}$ is either either clique or its complement. More precisely $\displaystyle\Gamma(E_{d_{i}})$ is $\displaystyle K_{\phi\left(\frac{n}{d_{i}}\right)}$ is $\displaystyle e_{i}^{2}$ divides $\displaystyle n$ otherwise it is $\displaystyle\overline{K}_{\phi\left(\frac{n}{d_{i}}\right)}$ (see [9]). Next, we state a results related to the unimodal and log-concave property of the independent domination polynomial of $\displaystyle\Gamma(\mathbb{Z}_{n})$ for several values of $\displaystyle n,$ proved as a main and interesting result in Gürsoy, Ülker and Gürsoy [13]. We restate it and avoid the ambiguity in the stated version of [13].

The proof of Theorem 10 from Gürsoy, Ülker and Gürsoy [13] reads the following:

The independent domination polynomials of $\displaystyle\mathbb{Z}_{pq},\mathbb{Z}_{p^{2}q},\mathbb{Z}_{pqr}$ and $\displaystyle\mathbb{Z}_{p^{\alpha}}$ are unimodal since the sequence of its coefficients $\displaystyle\alpha_{0},\alpha_{1},\dots,\alpha_{m}$ are a sequence of non-negative numbers, and these polynomials have only real zeros. Then, we have

from Newton’s inequalities. Therefore, the sequence is log-concave and unimodal.

1. 1.

   The zero of the independent domination polynomials of the zero divisor graphs of $\displaystyle\mathbb{Z}_{pq},\mathbb{Z}_{p^{2}q},\mathbb{Z}_{pqr},$ and $\displaystyle\mathbb{Z}_{p^{\alpha}}$ are not always real and they have complex zeros as well, where $\displaystyle p,q,r,$ are primes with $\displaystyle p>2$ and $\displaystyle\alpha$ is a positive integer. We will show it in next couple of examples along with their graphical representation on complex plane.

2. 2.

   Since the zeros of the independent domination polynomials of the zero divisor graphs of $\displaystyle\mathbb{Z}_{pq},\mathbb{Z}_{p^{2}q},\mathbb{Z}_{pqr}$ and $\displaystyle\mathbb{Z}_{p^{\alpha}}$ are not always real, so Newton’s inequalities are not applicable and log-concave cannot be established with this procedure used in the proof pf Theorem 10 of [13].

Next, we give a sequence of examples of the independent domination polynomials of the zero divisor graphs of the families of the commutative rings mentioned in Theorem 2.1 having complex zeros as well and are not unimodal. We consider the some examples from [13]. Their graphs are given in [13] and graphs of other examples can be drawn in a similar fashion.

1. 1.

   For $\displaystyle G\cong\Gamma(\mathbb{Z}_{pq})$ with $\displaystyle p=3$ and $\displaystyle q=5,$ the independent domination polynomial is $\displaystyle D_{i}(G,x)=x^{2}+x^{4}$ (see Theorem 4, [13]) and its zeros are $\displaystyle 0$ and $\displaystyle\pm i$. The graphical representation of the zeros of $\displaystyle D_{i}(G,x)$ is easy to visualise.

2. 2.

   For $\displaystyle G\cong\Gamma(\mathbb{Z}_{p^{2}q})$ with $\displaystyle p=5^{2}$ and $\displaystyle q=3,$ the independent domination polynomial is $\displaystyle D_{i}(G,x)=x^{10}+4x^{21}+x^{28}$ (see Example 1, [13]) and its set of zeros are

   	$\displaystyle\displaystyle\Big{\{}$	$\displaystyle\displaystyle 0,1.21378,-0.891122,-0.764225-0.955193i,-0.764225+0.955193i,-0.739951-0.469162i,$
   		$\displaystyle\displaystyle-0.739951+0.469162i,-0.358291-0.803288i,-0.358291+0.803288i,0.120955\,+0.88068i,$
   		$\displaystyle\displaystyle 0.120955\,-0.88068i,0.274622\,-1.18462i,0.274622\,+1.18462i,0.571168\,-0.662058i,$
   		$\displaystyle\displaystyle 0.571168\,+0.662058i,0.850704\,+0.241222i,0.850704\,-0.241222i,1.09747\,-0.533693i,$
   		$\displaystyle\displaystyle 1.09747\,+0.533693i\Big{\}}.$

   The graphical representation of the zeros of $\displaystyle D_{i}(G,x)$ is shown in Figure 1, where green dots represent the zeros.

   

   

   Zeros of $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{75}),x).$    Zeros of $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{105}),x).$

3. 3.

   For $\displaystyle G\cong\Gamma(\mathbb{Z}_{pqr})$ with $\displaystyle p=3,q=5$ and $\displaystyle r=7,$ the independent domination polynomial is $\displaystyle D_{i}(G,x)=x^{22}+x^{36}+x^{42}+x^{44}$ (see Example 2, [13]) and its set of zeros are

   	$\displaystyle\displaystyle\Big{\{}$	$\displaystyle\displaystyle 0,-0.937799-0.164137i,-0.937799+0.164137i,-0.907901-0.436924i,-0.907901+0.436924i,$
   		$\displaystyle\displaystyle-0.721752-0.635391i,-0.721752+0.635391i,-0.455901-0.860636i,-0.455901+0.860636i,$
   		$\displaystyle\displaystyle-0.29053-1.07456i,-0.29053+1.07456i,-1.i,0.\,+1.i,0.29053\,-1.07456i,0.29053\,+1.07456i,$
   		$\displaystyle\displaystyle 0.455901\,-0.860636i,0.455901\,+0.860636i,0.721752\,-0.635391i,0.721752\,+0.635391i,0.907901\,$
   		$\displaystyle\displaystyle-0.436924i,0.907901\,+0.436924i,0.937799\,-0.164137i,0.937799\,+0.164137i\Big{\}}.$

   The graphical representation of the zeros of $\displaystyle D_{i}(G,x)$ is shown in Figure 1 (right), where green dots represent the zeros.

4. 4.

   For $\displaystyle G\cong\Gamma(\mathbb{Z}_{p^{\alpha}})$ with $\displaystyle p=3$ and $\displaystyle\alpha=5,$ the independent domination polynomial is $\displaystyle D_{i}(G,x)=2x+6x^{55}+x^{72}$ (see Example 3, [13]) and the set of zeros of this polynomial is very long. So, we represent it by the graphical picture as shown in Figure 2

   

   

   Zeros of $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{243}),x).$    Zeros of $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{729}),x).$

5. 5.

   For $\displaystyle G\cong\Gamma(\mathbb{Z}_{p^{\alpha}})$ with $\displaystyle p=3$ and $\displaystyle\alpha=6,$ the independent domination polynomial is $\displaystyle D_{i}(G,x)=2x+6x^{163}+x^{216}+18x^{217}$ and the list of zeros of this polynomial is very long. So, we avoid listing them and present its graphical representation in Figure 2 (right).

Next, we will check the unimodal property of the independent domination polynomial of the zero divisor graphs for $\displaystyle\mathbb{Z}_{pq},\mathbb{Z}_{p^{2}q},\mathbb{Z}_{pqr}$ and $\displaystyle\mathbb{Z}_{p^{\alpha}}.$

We recall that the independent domination polynomial of $\displaystyle\Gamma(\mathbb{Z}_{p^{2}})$ is $\displaystyle(p-1)x,$ which is clearly unimodal as sequence of its coefficients form an increasing sequence of non-negative integers. From now onwards, whenever we consider $\displaystyle n=p^{\alpha}$, we assume $\displaystyle p>2$ to avoid $\displaystyle\Gamma(\mathbb{Z}_{p^{2}})$ case. Next for $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{2p}),x)=x+x^{p-1}$ to be unimodal, we have $\displaystyle a_{0}=0\leq a_{1}=1$ which must be less than or equal to coefficient of $\displaystyle x^{p-1}$, which can happen if and only if $\displaystyle p-1=2$ as its coefficient is already one. Otherwise if $\displaystyle p>2,$ then some coefficients between the linear term and $\displaystyle(p-1)$-th must be zero. Which will imply that $\displaystyle\eta(D_{i}(\Gamma(\mathbb{Z}_{2p}),x))>2,$ thereby contradicting the requirement for unimodal property. Also, by definition the log-concave property is satisfied trivially for $\displaystyle\Gamma(\mathbb{Z}_{p^{2}})$. However, for $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{2p}),x)=x+x^{p-1},$ we see that $\displaystyle 1=a_{1}^{2}\geq a_{0}a_{2}=0$ and for $\displaystyle i\geq 2$, we get $\displaystyle 0=a_{i}^{2}\geq a_{i-1}a_{i+1}=0$, which are always true, and it follows that $\displaystyle D_{i}(\Gamma(\mathbb{Z}_{2p}),x)$ is log-concave. We make these observation precise in the following result.

With a similar argument as in (ii) of Proposition 2.2, we have the following result.

We note that the number of oscillation in the independent domination polynomial of the above results is presciently three, since $\displaystyle a_{0}\leq\dots\leq a_{\phi(q)+\phi(pq)}\geq a_{\phi(q)+\phi(pq)+1}\geq\dots\geq a_{\phi(p^{2})}\leq a_{\phi(p^{2})+1}\geq a_{\phi(p^{2})+2}\dots\geq a_{\phi(p^{2})+\phi(pq)-1}\leq a_{\phi(p^{2})+\phi(pq)}.$

The following result gives the unimodal and log-concave property for the zero divisor graphs when $\displaystyle n$ is product of three primes and the result corrects Theorem 10 of [13] for $\displaystyle\Gamma(\mathbb{Z}_{pqr})$.

As classified by the above result, we mention some of the classes of the independent domination polynomials of zero divisor graphs of $\displaystyle\mathbb{Z}_{pqr}$ which are not log-concave.

1. 1.

   The independent domination polynomial of $\displaystyle G\cong\Gamma(\mathbb{Z}_{2\cdot 3\cdot 5})$ is $\displaystyle D_{i}(G,x)=x^{7}+x^{12}+x^{14}+x^{16},$ which is not log-concave, since $\displaystyle a_{15}^{2}=0\ngeq a_{14}\cdot a_{16}=1.$

2. 2.

   The independent domination polynomial of $\displaystyle G\cong\Gamma(\mathbb{Z}_{2\cdot 5\cdot 7})$ is $\displaystyle D_{i}(G,x)=x^{11}+x^{32}+x^{34}+x^{36},$ which is not log-concave, since $\displaystyle a_{35}^{2}=0\ngeq a_{34}\cdot a_{36}=1.$

3. 3.

   The independent domination polynomial of $\displaystyle G\cong\Gamma(\mathbb{Z}_{2\cdot 3\cdot 7})$ is $\displaystyle D_{i}(G,x)=x^{9}+x^{14}+x^{20}+x^{22},$ which is not log-concave, since $\displaystyle a_{21}^{2}=0\ngeq a_{20}\cdot a_{22}=1.$

4. 4.

   The independent domination polynomial of $\displaystyle G\cong\Gamma(\mathbb{Z}_{3\cdot 5\cdot 7})$ is $\displaystyle D_{i}(G,x)=x^{22}+x^{36}+x^{42}+x^{44},$ which is not log-concave, since $\displaystyle a_{43}^{2}=0\ngeq a_{42}\cdot a_{44}=1.$

For the above classes of polynomials, we see that $\displaystyle\phi(qr)+\phi(pr)+\phi(r)$ is bigger or sometimes $\displaystyle\phi(qr)+\phi(pr)+\phi(pq)$ is bigger, oscillation is exactly four, and they are not unimodal.

Next, we discuss the independent domination polynomial of $\displaystyle\Gamma(\mathbb{Z}_{n})$ when $\displaystyle n$ is a prime power. We will state the result without proof, since it is similar to Theorem 7 given in [13]. We recall number theoretic facts: $\displaystyle\sum_{i=1}^{\ell}p^{i}=p^{\ell}-1$, where $\displaystyle p$ is prime and $\displaystyle\ell$ is a positive integer and for relatively primes $\displaystyle p$ and $\displaystyle q$, we have $\displaystyle\phi(pq)=\phi(p)\phi(q)$.

The following results gives the non unimodal and log-concave property of $\displaystyle\Gamma(\mathbb{Z}_{n})$ when $\displaystyle n$ is prime power.

For $\displaystyle n=p^{2m},$ the independent domination polynomial of $\displaystyle G\cong\Gamma(\mathbb{Z}_{n})$ in standard form is