A note on "Extremal graphs with bounded vertex bipartiteness number"

TL;DR

This paper provides counterexamples to a previous theorem on extremal graphs with bounded vertex bipartiteness number, and offers a corrected theorem and proof to address the identified issues.

Contribution

It identifies errors in a prior theorem, supplies counterexamples, and presents a corrected version with a valid proof.

Findings

Two counterexamples disproving the original theorem

A corrected theorem with a valid proof

Clarification of the conditions for extremal graphs

Abstract

This paper is devoted to present two counterexamples to the theorem from \cite{MK} Maria R., Katherine T. M., Bernardo S. M., Extremal graphs with bounded vertex bipartiteness number, Linear Algebra Appl. 493 (2016) 28-36. Moreover, the corrected theorem and proof are presented.

Equations

$$\widehat{G}=K_{k}\vee\big{(}\overline{K_{\frac{n-k}{2}}}\vee\overline{K_{\frac{n-k}{2}}}\big{)}\in\Sigma_{k}(n).$$

$$\rho(\widehat{G})=\frac{1}{2}\Bigg{(}\frac{n+k-2}{2}+\sqrt{k^{2}+(3k-1)(n-k)+\Big{(}\frac{n-k+2}{2}\Big{)}^{2}}~{}\Bigg{)}.$$

$$\widehat{G}=K_{m}\vee\big{(}\overline{K_{\lfloor\frac{n-k}{2}\rfloor}}\vee\overline{K_{\lfloor\frac{n-k+1}{2}\rfloor}}\big{)}\in\Sigma_{k}(n).$$

$$\widetilde{\Upsilon}=\left(\begin{array}[]{ccc}k-1&s+1&s\\ k&0&s\\ k&s+1&0\\ \end{array}\right).$$

$$\sigma(A_{K_{4}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}})=\Big{\{}(-3)^{[1]},(-1)^{[3]},0^{[4]},(3-2\sqrt{6}~{})^{[1]},(3+2\sqrt{6}~{})^{[1]}\Big{\}}.$$

$$\displaystyle\rho(A_{K_{4}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}})$$

$$\sigma(A_{K_{5}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}})=\Big{\{}(-3)^{[1]},(-2)^{[1]},(-1)^{[4]},0^{[4]},9^{[1]}\Big{\}}.$$

$$\displaystyle\rho(A_{K_{5}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}})$$

$$\widehat{G}=K_{k}\vee\big{(}\overline{K_{\frac{n-k}{2}}}\vee\overline{K_{\frac{n-k}{2}}}\big{)}\in\Sigma_{k}(n).$$

$$\rho (A_{\hat{G}})=\frac{n+k-2+\sqrt{-7k^2+10kn-12k+n^2+4n+4}}{4}.$$

$$\widehat{G}=K_{k}\vee\big{(}\overline{K_{\lfloor\frac{n-k}{2}\rfloor}}\vee\overline{K_{\lfloor\frac{n-k+1}{2}\rfloor}}\big{)}\in\Sigma_{k}(n).$$

$$\widetilde{\Upsilon}=\left(\begin{array}[]{ccc}k-1&s+1&s\\ k&0&s\\ k&s+1&0\\ \end{array}\right).$$

$$\widetilde{\Upsilon}=\left(\begin{array}[]{ccc}k-1&r&s\\ k&0&s\\ k&r&0\\ \end{array}\right).$$

$$\left|\widetilde{{\rm Y}}-\lambda \cdot I\right|$$

$$=$$

$${\lambda }_1=\frac{k-n}{2},{\lambda }_{2,3}=\frac{n+k-2\pm \sqrt{-7k^2+10kn-12k+n^2+4n+4}}{4}.$$

$$\sigma(\widetilde{\Upsilon})=\Bigg{\{}\frac{k-n}{2},\frac{n+k-2\pm\sqrt{-7k^{2}+10kn-12k+n^{2}+4n+4}}{4}\Bigg{\}}.$$

$$\rho (\hat{G})=\frac{n+k-2+\sqrt{-7k^2+10kn-12k+n^2+4n+4}}{4}.$$

Full text

Keywords: Adjacency matrix,   Signless Laplacian matrix,    Maximal eigenvalue,   Vertex bipartiteness

1 Introduction

Throughout this paper we are concerned with finite undirected connected simple graphs. Let $G$ be a graph with vertices labelled $1,2,\dots,n$. The adjacency matrix $A_{G}$ of $G$ is an $n\times n$ matrix with the $(i,j)$-entry equal to 1 if vertices $i$ and $j$ are adjacent and 0 otherwise. The spectrum of a matrix $M$, denoted by $\sigma(M)$ is the multiplicities of the eigenvalues, which are represented in $\sigma(M)$ as powers in square brackets, e.g., $\sigma(M)=\{\lambda_{1}^{[m_{1}]},\lambda_{2}^{[m_{2}]},\cdots,\lambda_{q}^{[m_{q}]}\}$ indicates that $\lambda_{1}$ has multiplicity $m_{1}$, $\lambda_{2}$ has multiplicity $m_{2}$, and so on. The spectral radius of $M$ is $\rho(M)=max\{|\lambda|:\lambda\in\sigma(M)\}.$

We use $\overline{G}$ to denote the complement of $G$. The complete graph $K_{n}$ is a graph on $n$ vertices such that any two distinct vertices are connected. Let $K_{s,t}$ denote the complete bipartite graph whose partition classes have orders $s$ and $t.$ For two vertex-disjoint graphs $G_{1}$ and $G_{2}$, the join $G_{1}\vee G_{2}$ is the graph such that $V(G_{1}\vee G_{2})=V(G_{1})\cup V(G_{2})$ and $E(G_{1}\vee G_{2})=E(G_{1})\cup E(G_{2})\cup\{uv\mid u\in V(G_{1}),v\in V(G_{2})\}.$ For the underlying graph theoretical definitions and notations we follow [1, 2].

The fewest number of vertices whose deletion yields a bipartite graph from $G$ was defined by Fallat and Fan to be the vertex bipartiteness of $G$ and it is denoted by $v_{b}(G).$

Let $k$ be a natural number such that $3\leq n-k.$ The set $\Sigma_{k}(n)$ is defined by

$\{G=(V(G),E(G)):G$ is connected, $\mid V(G)\mid=n$ and $v_{b}(G)\leq k\}.$

The authors of [2] identified the graph in $\Sigma_{k}(n)$ with maximum spectral radius and maximum signless Laplacian spectral radius. They obtained the following theorem.

Theorem 1.1

(Theorem 3. in [2])* Let $1\leq k\leq n-3.$ Then the following hold.*

(a) If $n-k$ is even, then $\rho_{\widehat{G}}\geq\rho_{G}$ holds for all graphs $G\in\Sigma_{k}(n)$, where

[TABLE]

Equality holds if and only if $G=\widehat{G}.$ The expression for $\rho_{\widehat{G}}\geq\rho_{G}$ is given by

[TABLE]

(b) If $n-k$ is odd, then $\rho(\widehat{G})\geq\rho(G)$ holds for all graphs $G\in\Sigma_{k}(n)$, where

[TABLE]

Equality holds if and only if $G=\widehat{G}.$ The eigenvalue $\rho(\widehat{G})$ corresponds to the maximal eigenvalue of $\widetilde{\Upsilon}$ in

[TABLE]

However, for Theorem 1.1, we have the following counterexamples.

2 Some counterexamples

In this section, we propose two examples to show the bound of spectral radius is incorrect in Theorem 3 (a) of [2].

Example 2.1  Given a graph \widehat{G}=K_{4}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}, by routine calculations, we can obtain

[TABLE]

Then the spectral radius \rho(A_{K_{4}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}}) is $3+2\sqrt{6}$.

Note that $n=10,k=4$. Based on Theorem 1.1, one can get

[TABLE]

It is a contradiction.

Example 2.2  Given a graph \widehat{G}=K_{5}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}. Similarly, we can obtain

[TABLE]

Then the spectral radius \rho(A_{K_{5}\vee\big{(}\overline{K_{3}}\vee\overline{K_{3}}\big{)}}) is $9$.

Note that $n=11,k=5$. According to Theorem 1.1, one can arrive at

[TABLE]

which also arrives at a contradiction.

We present the corresponding theorem in next section, which corrects Theorem 1.1.

3 Main results

In this section, we first recall the known property of quotient matrix that will be used later.

Theorem 3.1

(See [3].) Suppose $\overline{M}$ is the quotient matrix of a partitioned symmetric matrix $M$, then the eigenvalues of $\overline{M}$ interlace the eigenvalues of $M.$ Moreover, if the interlacing is tight, then the partition of $M$ is regular. On the other hand, if the $M$ is regularly partitioned, then the eigenvalues of $\overline{M}$ are eigenvalues of $M$.

In what follows, we compute the eigenvalues of $\widehat{G}$ and obtain the following result.

Theorem 3.2

Let $1\leq k\leq n-3.$ Then the following hold.

(a) If $n-k$ is even, then $\rho(A_{\widehat{G}})\geq\rho(A_{G})$ holds for all graphs $G\in\Sigma_{k}(n)$, where

[TABLE]

Equality holds if and only if $G\cong\widehat{G}.$ The expression for $\rho(A_{\widehat{G}})$ is given by

[TABLE]

(b) If $n-k$ is odd, then $\rho(A_{\widehat{G}})\geq\rho(A_{G})$ holds for all graphs $G\in\Sigma_{k}(n)$, where

[TABLE]

Equality holds if and only if $G\cong\widehat{G}.$ The eigenvalue $\rho(A_{\widehat{G}})$ corresponds to the maximal eigenvalue of $\widetilde{\Upsilon}$ in

[TABLE]

Proof. By a suitable labelling, for \widehat{G}=K_{k}\vee\big{(}\overline{K_{\frac{n-k}{2}}}\vee\overline{K_{\frac{n-k}{2}}}\big{)}, the matrix $A_{\widehat{G}}$ has regular partitioning into blocks.

(a) If $n-k$ is even, $r=s=\frac{n-k}{2},$ then by Theorem 3.1, $\sigma(\widetilde{\Omega})\subseteq\sigma(A_{\widehat{G}})$, where

[TABLE]

Note that an irreducible nonnegative matrix, the only eigenvalue with a nonnegative eigenvector is its maximal eigenvalue (see [4]). If $M$ is an irreducible nonnegative matrix then the quotient matrix $\overline{M}$ is also irreducible and nonnegative. Consequently, the maximal eigenvalue of $\overline{M}$ also is the maximal eigenvalue of $M$.

We now compute the eigenvalues of $\widetilde{\Upsilon}$.

[TABLE]

We can obtain

[TABLE]

[TABLE]

Hence, the spectral radius $\rho(A_{\widehat{G}})$ is the maximal eigenvalue of $\sigma(\widetilde{\Upsilon})$. The expression for $\rho(A_{\widehat{G}})$ is given by

[TABLE]

For the proof of (b), readers can refer to the proof of Theorem 3 (b) [2], which is correct. Hence, we omit it here.

Acknowledgments

The work is partly supported by the National Science Foundation of China under Grant No. 11601006.