The energy and spectrum of non commuting graph

TL;DR

This paper investigates the spectral properties and energy of non-commuting graphs associated with dihedral groups and their products, providing explicit calculations and extending known results in algebraic graph theory.

Contribution

It computes the energy, Laplacian energy, and spectrum of non-commuting graphs for dihedral groups and their products, offering new explicit formulas and extending previous work.

Findings

Calculated the energy and spectrum of non-commuting graphs of D2n.

Extended results to non-commuting graphs of D2n × D2n and G × H.

Provided explicit formulas for energies of these graphs.

Abstract

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph {\Gamma}(G) of G is a graph with vertex set is non central elements of G and two vertices x, y are adjacent if and only if they are commute. In this paper we calculate the energy, Laplacian energy and spectrum of non-commuting graph of dihedral group D2n. Also we will obtain the energy of non-commuting graph of D2n \times D2n and G \times H, where G is a non-abelian finite group and H is an abelian finite group

Tables

Table 1. Table 1. The characteristic polynomial , eigenvalues and energy of some graphs.

groups	characteristic polynomial	eigenvalues	energy
$D_6$	$P_A\left(x\right)=\left(-x\right){\left(-x-1\right)}^2\left(x^2-2x-6\right)$	${\left(0\right)}_1,{\left(-1\right)}_2,\left(1+\sqrt{7}\right),\left(1-\sqrt{7}\right)$	$2+2\sqrt{7}$
$D_8$	$P_A\left(x\right)={\left(-x\right)}^3\left(-x+4\right){\left(-x-2\right)}^2$	${\left(0\right)}_3,{\left(4\right)}_1,{\left(-2\right)}_2$	$8$
$D_{10}$	$P_A\left(x\right)={\left(-x\right)}^3{\left(-x-1\right)}^4\left(x^2-4x-20\right)$	${\left(0\right)}_3,{\left(-1\right)}_4,\left(2-2\sqrt{6}\right),\left(2+2\sqrt{6}\right)$	$4+4\sqrt{6}$
$D_{12}$	$P_A\left(x\right)={\left(-x\right)}^6{\left(-x-2\right)}^2\left(x^2-4x-24\right)$	${\left(0\right)}_6,{\left(-2\right)}_2,\left(2+2\sqrt{7}\right),\left(2-2\sqrt{7}\right)$	$4+4\sqrt{7}$
$D_{16}$	$P_A\left(x\right)={\left(-x\right)}^9{\left(-x-2\right)}^3\left(x^2-6x-48\right)$	${\left(0\right)}_9,{\left(-2\right)}_3,\left(3+\sqrt{57}\right),\left(3-\sqrt{57}\right)$	$6+2\sqrt{57}$

Table 2. Table 2. The characteristic polynomial, eigenvalues and Laplacian energy of some graphs.

groups	characteristic polynomial	eigenvalues	energy
$D_8$	$P_A\left(x\right)=x{\left(x-4\right)}^3{\left(x-6\right)}^2$	${\left(0\right)}_1,{\left(4\right)}_3,{\left(6\right)}_2$	8
$D_{10}$	$P_A\left(x\right)=\left(-x\right){\left(x-5\right)}^3{\left(9-x\right)}^5$	${\left(0\right)}_1,{\left(5\right)}_3,{\left(9\right)}_5$	60
$D_{12}$	$P_A\left(x\right)=x{\left(x-6\right)}^3{\left(x-8\right)}_3{\left(x-10\right)}_3$	${\left(0\right)}_1,{\left(6\right)}_3,{\left(8\right)}_3,{\left(10\right)}_3$	$\frac{108}{5}$
$D_{14}$	$P_A\left(x\right)=\left(-x\right){\left(x-7\right)}^5{\left(13-x\right)}^7$	${\left(0\right)}_1,{\left(7\right)}_5,{\left(13\right)}_7$	126
$D_{16}$	$P_A\left(x\right)=x{\left(x-8\right)}^5{\left(x-12\right)}^4{\left(x-14\right)}^4$	${\left(0\right)}_1,{\left(8\right)}_5,{\left(12\right)}_4,{\left(14\right)}_4$	$\frac{304}{7}$

Equations

$$spec(\Gamma )=\left(\begin{array}{cccc}{\lambda }_1& {\lambda }_2& \dots & {\lambda }_t\\ k_1& k_2& \dots & k_t\end{array}\right).$$

$$E(\Gamma )=\sum _{i=1}^n\left|{\lambda }_i\right|,$$

$$LE(G)=\sum _{i=1}^n\left|{\mu }_i-\frac{2m}{n}\right|.$$

$$P(K_{n_1,n_2,\dots ,n_p},\lambda )={\lambda }^{(n-p)}(1-\sum _{i=1}^p\frac{n_i}{\lambda +n_i})\prod _{j=1}^p(\lambda +n_j).$$

$$E(K_{n_1,n_2,\dots ,n_p})=2{\lambda }_1.$$

$$P(K_{n_1,n_2},\lambda )={\lambda }^{n-2}({\lambda }^2-n_1{n}_2),$$

$${\lambda }_1(K{n}_1,n_2)=\sqrt{n_1{n}_2},E(K_{n_1,n_2})=2\sqrt{n_1{n}_2}.$$

$$P(K_{n_1,n_2,n_3},\lambda )={\lambda }^{(n-3)}({\lambda }^3-(n_1{n}_2+n_2{n}_3+n_3{n}_1)\lambda -2n_1{n}_2{n}_3).$$

$$spec({\Gamma }_{D_{2n}})=\left(\begin{array}{cccc}-2& 0& \frac{(n-2)-\sqrt{5n^2-12n+4}}{2}& \frac{(n-2)+\sqrt{5n^2-12n+4}}{2}\\ & & & \\ \frac{n}{2}-1& \frac{3n}{2}-3& 1& 1\end{array}\right).$$

$$spec({\Gamma }_{D_{2n}})=\left(\begin{array}{cccc}-1& 0& \frac{(n-1)-\sqrt{5n^2-6n+1}}{2}& \frac{(n-1)+\sqrt{5n^2-6n+1}}{2}\\ & & & \\ (n-1)& (n-2)& 1& 1\end{array}\right).$$

$$A_{ij}(\Gamma_{D_{2n}})=\left\{\begin{array}[]{rl}0&1\leq i,j\leq n-2\\ 0&i=k+t,j=k+s;t,s=0\ or\ 1\\ &and\ k=n-1,n+1,\ldots,2n-3\\ 1&o.w\end{array}\right.$$

$$A_{ij}(\Gamma_{D_{2n}})=\left\{\begin{array}[]{rl}0&1\leq i,j\leq n-1\\ 0&i=j=n,\ n+1,\ \ldots,\ 2n-1\\ 1&o.w\end{array}\right.$$

$$P_{\Gamma_{D_{2n}}}(x)=\left\{\begin{array}[]{rl}(-x)^{\frac{3n}{2}-3}(-x-2)^{\frac{n}{2}-1}(x^{2}-x(n-2)-n(n-2))&\text{if}\ \ n\ is\ even\\ (-x)^{n-2}(-x-1)^{n-1}(x^{2}-x(n-1)-n(n-1))&\text{if}\ \ n\ is\ odd.\end{array}\right.$$

$$E(\Gamma_{D_{2n}})=\left\{\begin{array}[]{rl}{(n-2)}+\sqrt{5n^{2}-12n+4}&\text{if}\ \ n\ is\ even\\ (n-1)+\sqrt{5n^{2}-6n+1}&\text{if}\ \ n\ is\ odd.\end{array}\right.$$

$$P_{{\Gamma }_G}(x)$$

$$+(-2q^6+6q^5-q^4-13q^3+15q^2-5q)x-(q-1{)}^4{q}^2(q-2)(q+1)]$$

$$(x+(q-1{)}^2{)}^q(x+q(q-1){)}^{\frac{q^2-q-2}{2}}(x+(q-1)(q-2){)}^{\frac{q^2+q-2}{2}},$$

$$G-Z(G)=({\dot{\bigcup }}_{g\in G}((PZ(G){)}^g-Z(G))\cup ({\dot{\bigcup }}_{g\in G}{I}^g-Z(G))\cup ({\dot{\bigcup }}_{g\in G}{D}^g-Z(G)),$$

$$t=\frac{q(q+1)}{2}+\frac{q(q-1)}{2}+q+1=q^2+q+1.$$

$$P_{{\Gamma }_G}(x)$$

$$\displaystyle\qquad+(-2q^{6}+6q^{5}-q^{4}-13q^{3}+15q^{2}-5q)x-(q-1)^{4}q^{2}(q-2)(q+1)\Big{]}$$

$$(x+(q-1{)}^2{)}^q(x+q(q-1){)}^{\frac{q^2-q-2}{2}}(x+(q-1)(q-2){)}^{\frac{q^2+q-2}{2}}.$$

$$E({\Gamma }_{GL(2,q)})=|{\gamma }_1|+|{\gamma }_2|+|{\gamma }_3|+q(q-1{)}^2+\frac{q^2-q-2}{2}q(q-1)+\frac{q^2+q-2}{2}(q-1)(q-2),$$

$$P_{{\Gamma }_G}(x)$$

$$+(-2q^6+6q^5-q^4-13q^3+15q^2-5q)x-(q-1{)}^4{q}^2(q-2)(q+1)]$$

$$(x+(q-1{)}^2{)}^q(x+q(q-1){)}^{\frac{q^2-q-2}{2}}(x+(q-1)(q-2){)}^{\frac{q^2+q-2}{2}}.$$

$$\Delta =(\frac{\alpha }{3}{)}^3+(\frac{\beta }{2}{)}^2.$$

$$E({\Gamma }_{GL(2,q)})$$

$$=|{\gamma }_1|+|{\gamma }_2|+|{\gamma }_3|+q\left|-(q-1{)}^2\right|+\frac{q^2-q-2}{2}\left|-q(q-1)\right|$$

$$+\frac{q^2+q-2}{2}\left|-(q-1)(q-2)\right|$$

$$=|{\gamma }_1|+|{\gamma }_2|+|{\gamma }_3|+q(q-1{)}^2+\frac{q^2-q-2}{2}q(q-1)$$

$$+\frac{q^2+q-2}{2}(q-1)(q-2).$$

$$spec(L({\Gamma }_{D_{2n}}))=\left(\begin{array}{cccc}2n-2& 2n-4& n& 0\\ \frac{n}{2}& \frac{n}{2}& n-3& 1\end{array}\right).$$

$$spec(L({\Gamma }_{D_{2n}}))=\left(\begin{array}{ccc}0& n& 2n-1\\ 1& n-2& n\end{array}\right).$$

$$P_{L(\Gamma_{D_{2n}})}{(x)}=\left\{\begin{array}[]{rl}x{(x-n)}^{(n-3)}{(x-{(2n-4)})}^{(\frac{n}{2})}{(x-{(2n-2)})}^{\frac{n}{2}}&\text{if}\ \ $n$\ is\ even\\ (-x)(n-x)^{n-2}((2n-1)-x)^{n}&\text{if}\ \ $n$\ is\ odd.\end{array}\right.$$

Full text

Key words: Non-commuting graph, Energy of a graph , Spectrum.

2010 Mathematics Subject Classification: 20D99, 05C50.

1. Introduction and preliminaries

Let $G$ be a non-abelian finite group and $Z(G)$ be its center. The non-commuting graph $\Gamma(G)$ of $G$ is a graph whose vertex set is $G-Z(G)$ and two vertices $x$ and $y$ are joined if and only if $xy\neq yx$. Note that if $G$ is an abelian, then $\Gamma(G)$ is the null graph.The non-commuting graph $\Gamma(G)$ was first considered by Paul Erdös, when he posed the following problem in 1975 $[7]$: Let $G$ be a group whose non-commuting graph has no infinite complete subgraph. Is it true that there is a finite bound on the cardinalities of complete subgraphs of $\Gamma(G)$? B. H. Neumann answered positively Erdös’s question in $[7]$. The adjacency matrix of graph $\Gamma$ is the $(0,1)$ matrix $A$ indexed by the vertex set $V(\Gamma)$ of $\Gamma$, where $A_{xy}=1$ when there is an edge from $x$ to $y$ in $\Gamma$ and $A_{xy}=0$ otherwise. The characteristic polynomial of $A$, denoted by $P_{A}(x)$, is the polynomial defined by $P_{A}(x)=det(xI-A)$ where $I$ denotes the identity matrix. The spectrum of a finite graph $\Gamma$ is by definition the spectrum of the adjacency matrix $A$, that is, its set of eigenvalues together with their multiplicities. Assume that $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{t}$ are $t$ distinct eigenvalues of $\Gamma$ with the corresponding multiplicities $k_{1},k_{2},\ldots,k_{t}$. We denote by

[TABLE]

Let $\Gamma$ be an undirected graph without loops. The Laplacian matrix of $\Gamma$ is the matrix $L$ indexed by the vertex set of $\Gamma$, with zero row sums. If $D$ is the diagonal matrix, indexed by the vertex set of $\Gamma$ such that $D_{xx}$ is the degree of $x$ then $L=D-A$. The energy of a graph $\Gamma$, denoted by $E(\Gamma)$, is defined as

[TABLE]

where $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}$ are the eigenvalues of the adjacency matrix of $\Gamma$. This concept was introduced by Gutman and is intensively studied in chemistry, since it can be used to approximate the total $\pi$-electron energy of a molecule(see, e.g.[3,4]).

Let $\Gamma$ be a graph with $n$ vertices and $m$ edges. Let $\mu_{1},\mu_{2},\ldots,\mu_{n}$ be the Laplacian eigenvalues of $\Gamma$. The Laplacian energy of a graph $\Gamma$, is defined as

[TABLE]

Theorem 1.1**.**

*([2,6]). Let $K_{n_{1},n_{2},\ldots,n_{p}}$ denote the complete $p$-partite graph,

$p\geq 1$, $n=n_{1}+n_{2}+\ldots+n_{p}$ and $n_{1}\geq n_{2}\geq\ldots\geq n_{p}>0$. Then*

[TABLE]

The spectrum of $K_{n_{1},n_{2},\ldots,n_{p}}$ consist of the spectral radius $\lambda_{1}$ determined from the equation $\sum_{i=1}^{p}\frac{n_{i}}{\lambda+n_{i}}=1$, eigenvalue [math] with multiplicity $n-p$ and $p-1$ eigenvalues situated in the intervals $[-n_{p},-n_{p-1}],\ldots,[-n_{2},-n_{1}]$.

Lemma 1.2**.**

*([5]) If $\lambda_{1}$ is the spectral redius of the complete multipartite graph $K_{n_{1},n_{2},\ldots,n_{p}}$, then

[TABLE]

Lemma 1.3**.**

([5]). $(1)$ If $p=2$, then

[TABLE]

[TABLE]

$(2)$* If $p=3$, then*

[TABLE]

2. The energy and Laplacian energy of non-commuting graphs of some special groups

In this section, we calculate the energy and Laplacian energy of non-commuting graph of dihedral group $D_{2n}$.

Theorem 2.1**.**

If $n$ is even and $n>4$, then

[TABLE]

If $n$ is odd, then

[TABLE]

If $n=4$, then $spec(\Gamma_{D_{2n}})=\begin{pmatrix}-2&0&4\\ 2&3&1\end{pmatrix}$.

Proof.

The adjacency matrix of $\Gamma_{D_{2n}}$ is equal to

[TABLE]

when n is even, and

[TABLE]

when $n$ is odd. By direct calculations

[TABLE]

This completes the proof. ∎

Corollary 2.2**.**

[TABLE]

In Table $1$, the energies of some non-commuting graphs of dihedral groups is given.

Theorem 2.3**.**

Let $G=GL{(2,q)}$, where $q=p^{n}>2$ (p is prime). Then

[TABLE]

where $t=q^{2}+q+1$.

Proof.

By [1] we have

[TABLE]

such that $D$ is the subgroup of all diagonal matrices in $G$, $I$ is a cyclic subgroup of $G$ of order $q^{2}-1$ and $P$ is the sylow $p$-subgroug of $G$ contaning all matrices as $\begin{bmatrix}1&x\\ 0&1\end{bmatrix}$. Moreover $\left|{D}\right|={(q-1)}^{2}$ and $\left|{PZ{(G)}}\right|=q(q-1)$. Also the number of conjugates of $D$, $I$ and $PZ(G)$ is equal to $\frac{q(q+1)}{2}$, $\frac{q(q-1)}{2}$ and $q+1$, respectively. Since $C_{G}(d)=D$ for any non-central element $d$ of $D$, $C_{G}(I)=I$ and $PZ(G)=C_{G}(x)$ for any non-trivial element of $P$, then the non-commuting graph of group $G$ is a complete $t$-partite graph where

[TABLE]

By Theorem 1.1,

[TABLE]

∎

Corollary 2.4**.**

[TABLE]

where $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ are roots of $f(x)=x^{3}+(-q^{4}+q^{3}+4q^{2}-6q+2)x^{2}+(-2q^{6}+6q^{5}-q^{4}-13q^{3}+15q^{2}-5q)x-(q-1)^{4}q^{2}(q-2)(q+1)$.

Proof.

By Theorem 2.3, we have

[TABLE]

Let $f(x)=x^{3}+(-q^{4}+q^{3}+4q^{2}-6q+2)x^{2}+(-2q^{6}+6q^{5}-q^{4}-13q^{3}+15q^{2}-5q)x-(q-1)^{4}q^{2}(q-2)(q+1)$, $b=(-q^{4}+q^{3}+4q^{2}-6q+2)$,

$c=(-2q^{6}+6q^{5}-q^{4}-13q^{3}+15q^{2}-5q)$ and $d=-(q-1)^{4}q^{2}(q-2)(q+1)$. Then we have $f(x)=x^{3}+bx^{2}+cx+d$. It is convenient to make the translation $x=y-\frac{b}{3}$, converting $f(x)$ into $g(y)=f(y-\frac{b}{3})=y^{3}+\alpha y+\beta$, where $\alpha=\frac{b^{2}-2b}{3}+c$ and $\beta=\frac{-b^{3}+3b^{2}-9bc}{27}+d$. We have

[TABLE]

Since $\Delta<0$, $g(y)$ has three real roots. Now let $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ be roots of $g(y)$. Then

[TABLE]

∎

Theorem 2.5**.**

If $n$ is even and $n>4$, then

[TABLE]

If $n$ is odd, then

[TABLE]

If $n=4$, then $spec(L(\Gamma_{D_{2n}}))=\begin{pmatrix}0&4&6\\ 1&3&2\end{pmatrix}$.

Proof.

By considering $L(\Gamma_{D_{2n}})$ and direct calculations, we have

[TABLE]

This completes the proof. ∎

Corollary 2.6**.**

[TABLE]

In Table $2$, the Laplacian characteristic polynomial and eigenvalues of non-commuting graphs of some dihedral groups is given.

3. The energy of non-commuting graph of direct product of groups

In this section, we calculate the energy of non-commuting graph of $D_{2n}\times D_{2n}$ and $G\times H$, which G is a non-abelian finite group and H is an abelian finite group.

Theorem 3.1**.**

Let $G$ be a non-abelian finite group and $H$ be an abelian group of order $n$. Then $E(\Gamma_{G\times H})=nE(\Gamma_{G}).$

Proof.

Let $h_{1},h_{2},\ldots,h_{n}$ be elements of $H$. Suppose $A(G)$ and $A(H)$ be adjacency matrices of non-commuting graph of groups $G$ and $H$, respectively. Then the adjacency matrix of $\Gamma_{G\times H}$ as the following form

[TABLE]

We have

[TABLE]

But

[TABLE]

Thus

[TABLE]

If $\lambda_{1},\lambda_{2},\ldots,\lambda_{t}$ be eigenvalues of $A(G)$, then

[TABLE]

are eigenvalues of $A(\Gamma_{G\times H})$. Therefore

[TABLE]

∎

Lemma 3.2**.**

Let $G=D_{8}$. Then

[TABLE]

and

[TABLE]

Proof.

For group $G$, we consider the adjacency matrix $A^{0}(G)$ similar to $A(G)$ by adding vertices $Z(G)$ to $A(G)$. Adjacency matrices of $A(G)$ and $A^{0}(G)$ differ only in some zero rows and zero columns. We have

[TABLE]

By direct calculations

[TABLE]

Thus

$\left|2A(G)^{0}-Ix-2J\right|^{2}=$

[TABLE]

$\hskip 91.04872pt=(-x)^{8}(-x-4)^{4}(x^{2}+8x-32)^{2}.$

Now assume that

[TABLE]

By calculating

[TABLE]

According to the description of the beginning of the proof, we conclude that

[TABLE]

and

[TABLE]

∎

Theorem 3.3**.**

Let $G=D_{2n}$. If $n$ is even and $n>4$, then

[TABLE]

*where $f(x)$ is a polynomial of degree $8$. Also the spectrum of $\Gamma_{G\times G}$ is equal to

$spec(\Gamma_{G\times G})=$*

[TABLE]

where $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, $\gamma_{4}$, $\gamma_{5}$, $\gamma_{6}$, $\gamma_{7}$, $\gamma_{8}$ and $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ are roots of $f(x)$ and \big{(}-x^{3}-(2n+4)x^{2}+16n(n-2)\big{)}, respectively.

Proof.

By the proof of Lemma 3.2, we have

[TABLE]

By direct calculations

[TABLE]

Thus

[TABLE]

Now assume that

[TABLE]

By direct calculations, we have

[TABLE]

[TABLE]

where $a=(n-2)^{2}$, $b=2(n-2)$ and $c=n(n-2)$. Therefore

[TABLE]

[TABLE]

Let

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $B^{T}$ is transpose of $B$. It follows that

[TABLE]

where $f(x)$ is a polynomial of degree $8$. This implies that

[TABLE]

Now let $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, $\gamma_{4}$, $\gamma_{5}$, $\gamma_{6}$, $\gamma_{7}$ and $\gamma_{8}$ be roots of $f(x)$ in $P_{\Gamma_{G\times G}}(x)$. Suppose $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ are roots of

\big{(}-x^{3}-(2n+4)x^{2}+16n(n-2)\big{)}. Then

$spec(\Gamma_{G\times G})=$

[TABLE]

∎

Corollary 3.4**.**

[TABLE]

Example 3.5**.**

[TABLE]

and

[TABLE]

*where $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\beta_{1}$, $\beta_{2}$, $\beta_{3}$ are roots of

$(x^{3}+16x^{2}-384)$ and $(x^{3}-384x+2304)$, respectively. Also $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, $\gamma_{4}$ are roots of $(x^{4}-104x^{3}-1152x^{2}+5376x+55296)$.*

Theorem 3.6**.**

*Let $G$ be a finite non-abelian group, where $\left|Z(G)\right|=t$. Then

[TABLE]

Proof.

By the proof of Lemma 3.2, the adjacency matrix of $\Gamma_{G\times S_{3}}$ as following

[TABLE]

Similar to the proof of Lemma 3.2, we have

[TABLE]

∎

Theorem 3.7**.**

Let $G$ be a finite non-abelian group, where $\left|Z(G)\right|=t$. If $n$ is even, then

[TABLE]

Proof.

By the proof of Lemma 3.2, we have

[TABLE]

By direct calculations

[TABLE]

∎

Theorem 3.8**.**

Let $G$ be a finite non-abelian group, where $\left|Z(G)\right|=t$. Then

[TABLE]

Proof.

It is similar to the proof of Lemma 3.2. ∎