Browder spectra of closed upper triangular operator matrices

Qingmei Bai; Alatancang Chen; Jingying Gao; Qingmei Bai; Alatancang Chen; Jingying Gao

doi:10.3934/math.2024248

AIMS Mathematics

2024, Volume 9, Issue 2: 5110-5121. doi: 10.3934/math.2024248

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Research article Special Issues

Browder spectra of closed upper triangular operator matrices

1.
School of Mathematical Sciences, Hohhot Minzu College, Hohhot, 010051, China
2.
Key Laboratory of Infinite-dimensional Hamiltonian System and Its Algorithm Application (Inner Mongolia Normal University), Ministry of Education, Hohhot, 010022, China
3.
Center for Applied Mathematics Inner Mongolia, Hohhot, 010022, China

Received: 10 September 2023 Revised: 16 January 2024 Accepted: 18 January 2024 Published: 25 January 2024
MSC : 47A25, 47A55

Let $T_{B} = \left[\begin{array}{ll} A & B \\ 0 & D \end{array}\right]$ be an unbounded upper operator matrix with diagonal domain, acting in $\mathcal H \oplus\mathcal K$ , where $\mathcal H$ and $\mathcal K$ are Hilbert spaces. In this paper, some sufficient and necessary conditions are characterized under which $T_{B}$ is Browder (resp., invertible) for some closable operator $B$ with $\mathcal D(B)\supset \mathcal D(D)$ . Further, a sufficient and necessary condition is given under which the Browder spectrum (resp., spectrum) of $T_{B}$ coincides with the union of the Browder spectrum (resp., spectrum) of its diagonal entries.

Keywords:

Citation: Qingmei Bai, Alatancang Chen, Jingying Gao. Browder spectra of closed upper triangular operator matrices[J]. AIMS Mathematics, 2024, 9(2): 5110-5121. doi: 10.3934/math.2024248

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Abstract

1. Introduction

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Gallian ^[3] has written a dynamic survey of graph labeling. MacDougall et al. ^[5] introduced the notion of a vertex magic total labeling of graphs. Let $G$ be a graph of order $n$ and size $m$ . A vertex magic total labeling of $G$ is defined as a one-to-one function

$f: V(G) \cup E(G) \rightarrow \{1, 2, \cdots, n+m\}$

with the property that for each vertex $u$ of $G$ ,

$f(u)+\sum\limits_{\substack{v\in N(u)}}f(uv) = k$

for some constant $k$ where $N(u)$ is the neighborhood of $u$ . The constant $k$ is called the magic constant for $f$ . The vertex-magic total labelings of wheels and related graphs were studied in ^[6], and later in ^[11]. The properties of the general graphs such as cycles, paths, complete graphs, wheels, bipartite graphs and trees, which satisfy the vertex magic total labelings, were studied in ^[10]. MacDougall et al. ^[4] introduced the concept of a super vertex magic total labeling. They defined a vertex magic total labeling to be super if

$f[V(G)] = \{1, 2, \cdots, n\}.$

In 2017, Nagaraj et al. ^[7] introduced the concept of an even vertex magic total labeling. They called a vertex magic total labeling as even if

$f[V(G)] = \{2, 4, \cdots, 2n\}.$

A graph $G$ is called an even vertex magic if there exists an even vertex magic total labeling of $G$ . We note that if $G$ is an even vertex magic, then $n \leq m$ . The following results, which appeared in ^[7], are useful to us.

Theorem 1.1. ^[7] Let $G$ be a nontrivial graph of order $n$ and size $m$ . If $G$ is an even vertex magic, then magic constant $k$ is given by the following:

${k = \frac{m^2+2mn+m}{n}}.$

A wheel $W_n$ , $n \geq 3$ , is a graph of order $n+1$ that contains a cycle $C_n$ , for which every vertex in the cycle $C_n$ is connected to one other vertex known as the hub. The edges of the wheel which are incident to the hub are called spokes. The vertices and edges of the cycle $C_n$ in $W_n$ are called rim vertices and rim edges, respectively. It was shown in ^[7] that a wheel $W_n$ has no even vertex magic total labeling, as we state next.

Theorem 1.2. ^[7] A wheel $W_n$ is not even vertex magic.

In this paper, the labeling problem is related to the work in ^[1]. In addition to the aforementioned vertex labeling by even numbers $2, 4, \cdots, 2n$ , they studied vertex labelings by using three consecutive numbers $0, 1, 2$ with some specific properties. These labelings were referred to as a weak Roman dominating function and a perfect Roman dominating function.

From the studies in ^[8,9,12], there exist graphs with the same order and size that are even vertex magics. Moreover, the wheel related graphs, namely fans, cycles and suns, having the even vertex magic total labelings were established in ^[7]. However, since these graphs have the same order and size, it is interesting and challenging to study wheel related graphs when the size is greater than the order, which have an even vertex magic total labeling.

The $t$ -fold wheel $W_{n, t}$ , $n \geq 3$ , $t \geq 1$ , is a wheel related graph derived from a wheel $W_n$ by duplicating the $t$ hubs, each adjacent to all rim vertices, and not adjacent to each other. It is observed that the $t$ -fold wheel $W_{n, t}$ has a size $nt+n$ that exceeds its order $n+t$ . The goal of this paper is to study conditions for an even vertex magic $W_{n, t}$ in terms of $n$ and $t$ . Furthermore, we also determine an even vertex magic total labeling of some $t$ -fold wheel $W_{n, t}$ .

2. The conditions for an even vertex magic $W_{n, t}$

Since the $1$ -fold wheel $W_{n, 1}$ is isomorphic to the wheel $W_n$ and by Theorem 1.1, $W_n$ is not an even vertex magic. In this section, we consider the $t$ -fold wheel $W_{n, t}$ , where $n$ and $t$ are integers with $n\geq 3$ and $t\geq2$ .

In order to present the conditions for an even vertex magic $W_{n, t}$ , we initially explore the magic constant of the $t$ -fold wheel $W_{n, t}$ of order $n+t$ and size $nt+n$ by employing Theorem 1.1.

Proposition 2.1. Let $n$ and $t$ be integers with $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then the magic constant is defined as follows:

$k = 2nt+3n+\frac{n^2t^2+2n^2t+n}{n+t}.$

We are able to show the bound of an integer $t$ for the $t$ -fold wheel having an even vertex magic total labeling as follows.

Proposition 2.2. Let $n$ and $t$ be integers with $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then $2 \leq t \leq n$ .

Proof. Suppose that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic with magic constant $k$ . By Proposition 2.1, we obtain the following:

$k = 2nt+3n+\frac{n^2t^2+2n^2t+n}{n+t}.$

On the contrary, assume that $t > n$ . Let $t = n+r$ , for some $r \geq 1$ . Then,

$n^2t^2+2n^2t+n = n^4+2n^3r+n^2r^2+2n^3+2n^2r+n$

and

$n+t = n+(n+r) = 2n+r.$

Let

$P(n) = n^4+2n^3r+n^2r^2+2n^3+2n^2r+n.$

By using the remainder theorem, the remainder when $P(n)$ is divided by $2n+r$ is as follows:

$P(-\frac{r}{2}) = \frac{r^4+4r^3-8r}{16}.$

$P(-\frac{r}{2}) = 0,$

then $r = -2$ , which is a contradiction. Thus,

$P(-\frac{r}{2}) \neq 0.$

Specifically, $n^2t^2+2n^2t+n$ is not divisible by $n+t$ . Thus, $k$ is not an integer, which is a contradiction. Therefore, $2 \leq t \leq n$ . □

According to Proposition 2.2, the $t$ -fold wheel $W_{3, t}$ is not an even vertex magic, where $t \geq 4$ . shows the even vertex magics $W_{3, 2}$ and $W_{3, 3}$ with magic constants $k = 36$ and $k = 50$ , respectively, where their vertices and edges are labeled by the even vertex magic total labelings. We present an even vertex magic total labeling of the $t$ -fold wheel $W_{n, t}$ by considering only the integer $n$ as the following results.

Figure 1. Even vertex magic total labelings of

$W_{3, 2}$ and

$W_{3, 3}$ .

DownLoad: Full-Size Img PowerPoint

Proposition 2.3. For every integer $n\geq 3$ , if the $n$ -fold wheel $W_{n, n}$ is an even vertex magic, then $n$ is odd.

Proof. Let $n$ be an integer with $n\geq 3$ . Suppose that the $n$ -fold wheel $W_{n, n}$ is an even vertex magic with a magic constant $k$ . On the contrary, assume that $n$ is even. There exists an integer $q$ such that $n = 2q$ . By Proposition 2.1,

$k = 2n^2+3n+\frac{n^3+2n^2+1}{2}.$

Since

$n^3+2n^2+1 = 8q^3+8q^2+1$

is odd, $n^3+2n^2+1$ is not divisible by $2$ . Thus, $k$ is not an integer, which is a contradiction. Therefore, $n$ is odd. □

As we have seen in , the $3$ -fold wheel $W_{3, 3}$ is an even vertex magic, as indicated by Proposition 2.3. By an argument similar to the one used in the proof of Proposition 2.3, we obtain the condition for an even vertex magic $W_{n, n-2}$ , as we now show.

Proposition 2.4. For every integer $n\geq 4$ , if the $(n-2)$ -fold wheel $W_{n, n-2}$ is an even vertex magic, then $n$ is even.

The even vertex magic total labeling of the $2$ -fold wheel $W_{4, 2}$ with a magic constant $k = 50$ is shown in Figure 2.

Figure 2. Even vertex magic total labeling of

$W_{4, 2}$ .

DownLoad: Full-Size Img PowerPoint

In order to deduce an even vertex magic total labeling of the $t$ -fold wheel for achieving the main result, we need some additional notation for the $t$ -fold wheel $W_{n, t}$ . For every pair of integers $n\geq 3$ and $t\geq2$ , let

$V(W_{n, t}) = \{u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_t\}$

and

$E(W_{n, t}) = \{u_iu_{i+1} | 1 \leq i \leq n-1\}\cup\{u_nu_1\}\cup\{u_iv_j | 1 \leq i \leq n, 1 \leq j \leq t\}.$

Suppose the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. Then, for any even vertex magic total labeling $f$ of $W_{n, t}$ , let

$S_{rv} = \sum\limits_{i = 1}^{n}f(u_i), \ \ \ S_{re} = \sum\limits_{i = 1}^{n-1}f(u_iu_{i+1})+f(u_nu_1)$

and

$S_{h} = \sum\limits_{j = 1}^{t}f(v_j), \ \ \ S_{s} = \sum\limits_{j = 1}^{t} \sum\limits_{i = 1}^{n}f(u_iv_j).$

Next, we present the following lemma to show the necessary condition for an even vertex magic $W_{n, t}$ with the following magic constant:

$k = 2nt+3n+\frac{n^2t^2+2n^2t+n}{n+t}.$

Note that

$S_{rv}+2S_{re}-S_{h} = (n-t)k.$

Lemma 2.5. Let $n$ and $t$ be integers where $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then

$S_{rv}+2S_{re}-S_{h} = \\ (t^2+4t+3)n^2+(-2t^3-6t^2-3t+1)n+\frac{(t^3+2t^2-1)(2nt)}{n+t}.$

With the aid of Lemma 2.5 and Proposition 2.2, the necessary condition for an even vertex magic total labeling of the $t$ -fold wheel $W_{n, t}$ can also be given in terms of $n$ and $t$ .

Proposition 2.6. Let $n$ and $t$ be integers where $n\geq 3$ and $t\geq2$ . If the $t$ -fold wheel $W_{n, t}$ is an even vertex magic, then

$(-t^2-2t+1)n^2+(2t^3+6t^2+7t+1)n-(t^2+t)-\frac{(t^3+2t^2-1)(2nt)}{n+t} \geq 0.$

Proof. Suppose that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. By Lemma 2.5,

$S_{rv}+2S_{re}-S_{h} = (t^2+4t+3)n^2+(-2t^3-6t^2-3t+1)n+\frac{(t^3+2t^2-1)(2nt)}{n+t}.$

Next, we consider the maximum of $(S_{rv}+2S_{re}-S_{h})$ .

By Proposition 2.2, $2\leq t \leq n$ , and then $2n+2t < nt+n+t+1$ . The maximum of

$\begin{array}{l} (S_{rv}+2S_{re}-S_{h}) & = & \sum\limits_{\substack{2t+2\leq i\leq 2n+2t\\ i\; \text{is even}}}i+2\sum\limits_{i = nt+n+t+1}^{nt+2n+t}i-\sum\limits_{\substack{2\leq i\leq 2t\\ i\; \text{is even}}}i \\ & = & \left(\sum\limits_{\substack{2\leq i\leq 2n+2t\\ i\; \text{is even}}}i-\sum\limits_{\substack{2\leq i\leq 2t\\ i\; \text{is even}}}i\right)+2\left(\sum\limits_{i = 1}^{nt+2n+t}i-\sum\limits_{i = 1}^{nt+n+t}i\right)-\sum\limits_{\substack{2\leq i\leq 2t\\ i\; \text{is even}}}i \\ & = & \left(\frac{(2n+2t)(2n+2t+2)}{4}-\frac{(2t)(2t+2)}{4}\right) \\ && +2\left(\frac{(nt+2n+t)(nt+2n+t+1)}{2}-\frac{(nt+n+t)(nt+n+t+1)}{2}\right) -\frac{(2t)(2t+2)}{4} \\ & = & 2n^2t+4n^2+4nt-t^2+2n-t \, . \end{array}$

Since $S_{rv}+2S_{re}-S_{h}$ does not exceed the maximum of $(S_{rv}+2S_{re}-S_{h})$ , the maximum of

$(S_{rv}+2S_{re}-S_{h}) - (S_{rv}+2S_{re}-S_{h}) \geq 0.$

Therefore,

$(-t^2-2t+1)n^2+(2t^3+6t^2+7t+1)n-(t^2+t)-\frac{(t^3+2t^2-1)(2nt)}{n+t} \geq 0.$

□

Now, we investigate the sufficient condition for a labeling $f$ that can be an even vertex magic total labeling of $W_{n, n}$ when $n$ is odd.

Theorem 2.7. Let $n$ be an odd integer where $n \geq 3$ . For every $n$ -fold wheel $W_{n, n}$ , let

$f: V(W_{n, n}) \cup E(W_{n, n}) \rightarrow \{1, 2, \ldots, n^2+3n\}$

be defined by the following:

$\begin{array}{l} f(u_i) ~ = ~ 2i, ~~ \mathit{\mbox{if }}~~ 1 \leq i \leq n , \\ f(v_j) ~ = ~ 2n+2j, ~~ \mathit{\mbox{if }}~~ 1 \leq j \leq n , \\ f(u_iu_{i+1}) ~ = ~ 2n+1-2i, ~~ \mathit{\mbox{if}}~~ 1 \leq i \leq n-1 , \\ f(u_nu_1) ~ = ~ 1, ~~ \\ f(u_{n+1-j}v_j) ~ = ~ n^2+3n+1-2j, ~~ \mathit{\mbox{if }}~~ 1 \leq j \leq n , \\ f[EH]-f[\{u_{n+1-j}v_j| 1 \leq j \leq n\}] ~ = ~ \{2n+1, 2n+3, \ldots, n^2+n-1\} ~~ \\ ~~ \cup\{4n+2, 4n+4, \ldots, n^2+3n\}, ~~ \mathit{\mbox{if}}~~ EH = \{u_iv_j| 1 \leq i, j \leq n\} \, . \end{array}$

$\sum\limits_{j = 1}^{n-1}f(u_1v_j) = \frac{n^3+4n^2-5}{2},$

then $f$ can be an even vertex magic total labeling of $W_{n, n}$ .

Proof. Assume that

$\sum\limits_{j = 1}^{n-1}f(u_1v_j) = \frac{n^3+4n^2-5}{2}.$

We have that

$\begin{array}{l} S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j) & = & \sum\limits_{\substack{2n+1\leq i\leq n^2+n-1 \\ i\; \text{is odd}}}i + \sum\limits_{\substack{4n+2\leq i\leq n^2+3n \\ i\; \text{is even}}}i \\ & = & \left(\sum\limits_{\substack{1\leq i\leq n^2+n-1\\ i\; \text{is odd}}}i - \sum\limits_{\substack{1\leq i\leq 2n-1\\ i\; \text{is odd}}}i\right) + \left(\sum\limits_{\substack{2\leq i\leq n^2+3n\\ i\; \text{is even}}}i - \sum\limits_{\substack{2\leq i\leq 4n\\ i\; \text{is even}}}i\right) \\ & = & \left(\frac{(n^2+n)^2}{4} - \frac{(2n)^2}{4}\right) + \left(\frac{(n^2+3n)(n^2+3n+2)}{4} - \frac{4n(4n+2)}{4}\right) \\ & = & \frac{n^4+4n^3-4n^2-n}{2}, \end{array}$

and then,

$\left(S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j)\right)-\sum\limits_{j = 1}^{n-1}f(u_1v_j) = \frac{n^4+3n^3-8n^2-n+5}{2}\, .$

Next, we consider the sum of the label of each vertex and the labels of all edges incident to this vertex. By the assumption, for $1 \leq j\leq n$ ,

$\begin{array}{l} f(v_j)+\sum\limits_{i = 1}^{n}f(u_iv_j) & = & f(v_j)+f(u_{n+1-j}v_j)+\sum\limits_{\substack{1\leq i\leq n\\ i \neq n+1-j}}f(u_iv_j) \\ & = & f(v_j)+f(u_{n+1-j}v_j)+\frac{S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j)}{n} \\ & = & (2n+2j)+(n^2+3n+1-2j)+\frac{n^3+4n^2-4n-1}{2}\\ & = & \frac{n^3+6n^2+6n+1}{2}, \end{array}$

$\begin{array}{l} f(u_1)+f(u_1u_2)&+&f(u_nu_1)+\sum\limits_{j = 1}^{n}f(u_1v_j)\\ & = & f(u_1)+f(u_1u_2)+f(u_nu_1)+f(u_1v_n)+\sum\limits_{j = 1}^{n-1}f(u_1v_j)\\ & = & 2+(2n+1-2)+1+(n^2+n-1+2)+\frac{n^3+4n^2-5}{2} \\ & = & \frac{n^3+6n^2+6n+1}{2} \, . \end{array}$

For $2 \leq i\leq n,$

$\begin{array}{l} f(u_i)+f(u_iu_{i+1})&+&f(u_{i-1}u_i)+\sum\limits_{j = 1}^{n}f(u_iv_j)\\ & = & f(u_i)+f(u_iu_{i+1})+f(u_{i-1}u_i)+f(u_iv_{n+1-i})+\sum\limits_{\substack{1\leq j\leq n\\ j \neq n+1-i}}f(u_iv_j)\\ & = & f(u_i)+f(u_iu_{i+1})+f(u_{i-1}u_i)+f(u_iv_{n+1-i})+\frac{ \left(S_{s}-\sum\limits_{j = 1}^{n}f(u_{n+1-j}v_j)\right)-\sum\limits_{j = 1}^{n-1}f(u_1v_j)}{n-1}\\ & = & 2i+(2n+1-2i)+(2n+1-2i+2)+(n^2+n-1+2i)+\frac{n^3+4n^2-4n-5}{2}\\ & = & \frac{n^3+6n^2+6n+1}{2} \, . \end{array}$

Therefore, $f$ can be an even vertex magic total labeling of $W_{n, n}$ with a magic constant

$k = \frac{n^3+6n^2+6n+1}{2}.$

□

Now, we investigate the sufficient condition for a labeling $f$ that can be an even vertex magic total labeling of $W_{n, n-2}$ when $n$ is even.

Theorem 2.8. Let $n$ be an even integer with $n \geq 4$ . For every $(n-2)$ -fold wheel $W_{n, n-2}$ , let

$f: V(W_{n, n-2}) \cup E(W_{n, n-2}) \rightarrow \{1, 2, \ldots, n^2+n-2\}$

be defined by the following:

$\begin{array}{rcll} f(u_i) & = & 2i, &\mathit{\mbox{if }}~~ 1 \leq i \leq n , \\ f(v_j) & = & 2n+2j, &\mathit{\mbox{if}}~~ 1 \leq j \leq n-2 , \\ f[EC] & = & \{a_1, a_2, \cdots, a_n\}, & \mathit{\mbox{if}}~~ EC = \{u_iu_{i+1}, u_nu_1 | 1 \leq i \leq n-1 \}, \\ f[EH] & = & \{1, 2, \cdots, n^2+n-2\} &\\&& -\{2, 4, \cdots, 2n+4, a_1, a_2, \cdots, a_n\}, &\mathit{\mbox{if}}~~ EH = \{u_iv_j| 1 \leq i \leq n, 1 \leq j \leq n-2\} . \end{array}$

$S_{s} = \frac{n^4+n^3-15n^2+20n-4}{2},$

then $f$ can be an even vertex magic total labeling of $W_{n, n-2}$ .

Proof. Assume that

$S_{s} = \frac{n^4+n^3-15n^2+20n-4}{2}.$

It suffices to show that for each vertex $u$ of $W_{n, n-2}$ ,

$f(u)+\sum\limits_{\substack{v\in N(u)}}f(uv) = k,$

where

$k = \frac{n^3+3n^2-3n}{2}.$

To do this, we consider the relevant sums, as follows.

Since the sum of the labels of all rim edges is equal to the sum of the labels of all vertices and the labels of all edges subtracted by the sum of the labels of all vertices and the labels of all spokes, it follows that

$\begin{array}{l} S_{re} & = & \sum\limits_{i = 1}^{n^2+n-2}i - \sum\limits_{\substack{2\leq i\leq 4n-4 \\ i\; \text{is even}}}i -S_{s}\\ & = & \frac{(n^2+n-2)(n^2+n-1)}{2} - \frac{(4n-4)(4n-2)}{4}-\frac{n^4+n^3-15n^2+20n-4}{2} \\ & = & \frac{n^3+5n^2-11n+2}{2}. \end{array}$

Since the sum of the labels of all hubs is equal to the sum of even integers from $2n+2$ to $4n-4$ ,

$\begin{array}{l} S_{h}+S_{s} & = & \sum\limits_{\substack{2n+2\leq i\leq 4n-4 \\ i\; \text{is even}}}i +S_{s}\\ & = & \left(\sum\limits_{\substack{2\leq i\leq 4n-4 \\ i\; \text{is even}}}i-\sum\limits_{\substack{2\leq i\leq 2n \\ i\; \text{is even}}}i\right) +S_{s}\\ & = & \frac{(4n-4)(4n-2)}{4} - \frac{(2n)(2n+2)}{4}+\frac{n^4+n^3-15n^2+20n-4}{2}\\ & = & \frac{n^4+n^3-9n^2+6n}{2}. \end{array}$

Since the sum of the labels of all rim vertices is equal to the sum of even integers from $2$ to $2n$ ,

$\begin{array}{l} S_{rv}+2S_{re}+S_{s} & = & \sum\limits_{\substack{2\leq i\leq 2n \\ i\; \text{is even}}}i+2S_{re}+S_{s} \\ & = &\frac{(2n)(2n+2)}{4}+2\left(\frac{n^3+5n^2-11n+2}{2}\right)+\frac{n^4+n^3-15n^2+20n-4}{2}\\ & = & \frac{n^4+3n^3-3n^2}{2} . \end{array}$

Next, we consider the sum of the label of each vertex and the labels of all edges incident to this vertex. We have the sum of the label of each hub and the labels of all edges incident to this hub as follows.

For $1 \leq j\leq n-2$ ,

$f(v_j)+\sum\limits_{i = 1}^{n}f(u_iv_j) = \frac{S_{h}+S_{s}}{n-2} = \frac{n^3+3n^2-3n}{2}.$

We obtain the sum of the label of each rim vertex and the labels of all edges incident to this rim vertex as follows.

For $2 \leq i\leq n-1,$

$f(u_i)+f(u_iu_{i+1})+f(u_{i-1}u_i)+\sum\limits_{j = 1}^{n-2}f(u_iv_j) = \frac{ S_{rv}+2S_{re}+S_{s}}{n} = \frac{n^3+3n^2-3n}{2}.$

Similarly,

$f(u_n)+f(u_nu_1)+f(u_{n-1}u_n)+\sum\limits_{j = 1}^{n-2}f(u_nv_j) = \frac{n^3+3n^2-3n}{2}$

and

$f(u_1)+f(u_1u_2)+f(u_nu_1)+\sum\limits_{j = 1}^{n-2}f(u_1v_j) = \frac{n^3+3n^2-3n}{2}.$

Therefore, $f$ can be an even vertex magic total labeling of $W_{n, n-2}$ with the following magic constant:

$k = \frac{n^3+3n^2-3n}{2}.$

□

3. An even vertex magic $W_{n, t}$ where $3\leq n\leq 9$

In this section, we establish a characterization of an even vertex magic $W_{n, t}$ for an integer $3\leq n \leq 9$ . First, we present an $n$ -fold wheel $W_{n, n}$ which has an even vertex magic total labeling for every odd integer $3\leq n \leq 9$ as follows.

Theorem 3.1. For every odd integer $3\leq n \leq 9$ , the $n$ -fold wheel $W_{n, n}$ is an even vertex magic.

Proof. Let $n$ be an odd integer where $3\leq n \leq 9$ . We define

$f: V(W_{n, n}) \cup E(W_{n, n}) \rightarrow \{1, 2, \cdots, n^2+3n\},$

as the sufficient condition of Theorem 2.7, by

$\begin{array}{rcll} f(u_i) & = & 2i, &\mbox{if }~~ 1 \leq i \leq n , \\ f(v_j) & = & 2n+2j, &\mbox{if}~~ 1 \leq j \leq n , \\ f(u_iu_{i+1}) & = & 2n+1-2i, &\mbox{if}~~ 1 \leq i \leq n-1 , \\ f(u_nu_1) & = & 1, & \end{array}$

and for $1 \leq i, j \leq n$ , $f(u_iv_j)$ are shown in Tables 1–4,

Table 1. Labels of edges

$u_iv_j$ of

$W_{3, 3}$ by

$f$ , for

$1 \leq i, j \leq 3$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$
$u_1$	11	18	13
$u_2$	14	15	9
$u_3$	17	7	16

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Table 2. Labels of edges

$u_iv_j$ of

$W_{5, 5}$ by

$f$ , for

$1 \leq i, j \leq 5$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$
$u_1$	15	21	36	38	31
$u_2$	13	30	29	33	28
$u_3$	40	24	35	19	17
$u_4$	34	37	11	23	32
$u_5$	39	27	26	22	25

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Table 3. Labels of edges

$u_iv_j$ of

$W_{7, 7}$ by

$f$ , for

$1 \leq i, j \leq 7$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$	$v_7$
$u_1$	15	51	64	40	29	68	57
$u_2$	66	43	17	38	19	59	70
$u_3$	25	45	56	48	61	32	47
$u_4$	52	31	33	63	54	53	30
$u_5$	42	35	65	34	60	36	46
$u_6$	55	67	23	58	49	27	41
$u_7$	69	50	62	37	44	39	21

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Table 4. Labels of edges

$u_iv_j$ of

$W_{9, 9}$ by

$f$ , for

$1 \leq i, j \leq 9$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$	$v_7$	$v_8$	$v_9$
$u_1$	73	33	102	69	19	79	63	86	91
$u_2$	43	56	35	62	49	89	85	93	87
$u_3$	75	61	48	53	104	92	95	23	50
$u_4$	80	58	72	25	108	97	77	47	39
$u_5$	51	96	59	81	99	37	55	57	70
$u_6$	60	100	44	101	67	40	41	90	64
$u_7$	38	83	103	82	74	78	29	76	46
$u_8$	88	105	94	65	42	66	52	31	68
$u_9$	107	21	54	71	45	27	106	98	84

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For every odd integer $3 \leq n\leq 9$ , the labeling $f$ , as defined above, is an even vertex magic total labeling of the $n$ -fold wheel $W_{n, n}$ with magic constants $k = 50, 153, 340$ and $635$ , respectively. Therefore, $W_{n, n}$ is an even vertex magic. □

As a consequence of an even vertex magic $W_{3, 2}$ , Proposition 2.2 and Theorem 3.1, in any $t$ -fold wheel $W_{3, t}$ , we are able to show that both $W_{3, t}$ and $W_{3, t}$ are only even vertex magics.

Theorem 3.2. For every integer $t \geq 2$ , the $t$ -fold wheel $W_{3, t}$ is an even vertex magic if and only if $t = 2, 3$ .

The following result gives the necessary and sufficient condition for the $t$ -fold wheel $W_{n, t}$ to be an even vertex magic for every odd integer $5\leq n \leq 9$ .

Theorem 3.3. For every odd integer $5\leq n \leq 9$ and an integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic if and only if $t = n$ .

Proof. Let $n$ be an odd integer where $5\leq n \leq 9$ and $t$ is an integer where $t \geq 2$ . Assume that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. By Proposition 2.2, $2\leq t \leq n$ .

Case 1. $n = 5, 7.$ If $2\leq t \leq n-1$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. Therefore, $t = n$ .

Case 2. $n = 9.$ If either $t = 2$ or $4\leq t \leq n-1$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. If $t = 3$ , then,

$2nt^3-n^2t^2-2n^2t+6nt^2+7nt+n^2-t^2+n-t-\frac{2nt^4+4nt^3-2nt}{n+t} = -174 < 0,$

which is a contradiction with Proposition 2.6. Therefore, $t = n$ .

Conversely, assume $t = n$ . By Theorem 3.1, $W_{n, t}$ is an even vertex magic. □

We show an even vertex magic total labeling of $W_{n, n-2}$ for every even integer $4\leq n \leq 8$ as follows.

Theorem 3.4. For every even integer $4 \leq n \leq 8$ , the $(n-2)$ -fold wheel $W_{n, n-2}$ is an even vertex magic.

Proof. Let $n$ be an even integer with $4 \leq n \leq 8$ . We define

$f: V(W_{n, n-2}) \cup E(W_{n, n-2}) \rightarrow \{1, 2, \ldots, n^2+n-2\}$

as the sufficient condition of Theorem 2.8, by

$\begin{array}{rcll} f(u_i) & = & 2i, &\mbox{if}~~ 1 \leq i \leq n , \\ f(v_j) & = & 2n+2j, &\mbox{if}~~ 1 \leq j \leq n-2 , \end{array}$

for $1 \leq i\leq n-1$ , $f(u_iu_{i+1})$ and $f(u_nu_1)$ are shown in Tables 5–7.

Table 5. Labels of edges

$u_iu_{i+1}$ and

$u_4u_1$ of

$W_{4, 2}$ by

$f$ , for

$1 \leq i\leq 3$ .

$f(u_1u_2)$	$f(u_2u_3)$	$f(u_3u_4)$	$f(u_4u_1)$
18	9	11	13

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Table 6. Labels of edges

$u_iu_{i+1}$ and

$u_6u_1$ of

$W_{6, 4}$ by

$f$ , for

$1 \leq i\leq 5$ .

$f(u_1u_2)$	$f(u_2u_3)$	$f(u_3u_4)$	$f(u_4u_5)$	$f(u_5u_6)$	$f(u_6u_1)$
40	39	38	13	17	19

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Table 7. Labels of edges

$u_iu_{i+1}$ and

$u_8u_1$ of

$W_{8, 6}$ by

$f$ , for

$1 \leq i\leq 7$ .

$f(u_1u_2)$	$f(u_2u_3)$	$f(u_3u_4)$	$f(u_4u_5)$	$f(u_5u_6)$	$f(u_6u_7)$	$f(u_7u_8)$	$f(u_8u_1)$
70	68	66	64	62	19	13	11

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And for $1 \leq i\leq n$ and $1 \leq j\leq n-2$ , $f(u_iv_j)$ are shown in Tables 8–10.

Table 8. Labels of edges

$u_iv_j$ of

$W_{4, 2}$ by

$f$ , for

$1 \leq i\leq 4$ and

$1 \leq j\leq 2$ .

$f(u_iv_j)$	$v_1$	$v_2$
$u_1$	16	1
$u_2$	14	5
$u_3$	7	17
$u_4$	3	15

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Table 9. Labels of edges

$u_iv_j$ of

$W_{6, 4}$ by

$f$ , for

$1 \leq i\leq 6$ and

$1 \leq j\leq 4$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$
$u_1$	36	24	3	29
$u_2$	11	25	33	1
$u_3$	26	9	5	30
$u_4$	22	23	34	15
$u_5$	37	21	28	27
$u_6$	7	35	32	31

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Table 10. Labels of edges

$u_iv_j$ of

$W_{8, 6}$ by

$f$ , for

$1 \leq i\leq 8$ and

$1 \leq j\leq 6$ .

$f(u_iv_j)$	$v_1$	$v_2$	$v_3$	$v_4$	$v_5$	$v_6$
$u_1$	69	3	40	56	39	50
$u_2$	1	67	5	59	31	35
$u_3$	21	17	65	7	44	46
$u_4$	42	38	23	33	9	57
$u_5$	61	27	37	34	30	15
$u_6$	41	53	49	32	47	25
$u_7$	29	60	54	52	63	36
$u_8$	58	55	45	43	51	48

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For every even integer $4 \leq n \leq 8$ , the labeling $f$ , as defined above, is an even vertex magic total labeling of the $(n-2)$ -fold wheel $W_{n, n-2}$ with magic constants $k = 50, 153$ and $340$ , respectively. Therefore, $W_{n, n-2}$ is an even vertex magic. □

There is a similar methodology of the proof of Theorem 3.4, which is also used in the study of graph operations (see ^[2]). Next, we determine a characterization of the $t$ -fold wheel $W_{n, t}$ to be an even vertex magic for every even integer $4 \leq n \leq 8$ . In order to we need to present the following lemma involving a $3$ -fold wheel $W_{8, 3}$ .

Lemma 3.5. The $3$ -fold wheel $W_{8, 3}$ is not an even vertex magic.

Proof. On the contrary, assume that the $3$ -fold wheel $W_{8, 3}$ is an even vertex magic with a magic constant $k$ . Since $W_{8, 3}$ has an order $11$ and a size $32$ and by Proposition 2.1, $k = 160$ . We have that

$S_{rv} = 132, \ \ 2S_{re} = 1, 628-2S_{s}$

and

$S_{rv}+2S_{re}+S_{s} = 8k = 1, 280.$

Thus, $S_{s} = 480$ . However, $S_{h}+S_{s} = 3k = 480$ . This is a contradiction because $S_{h} > 0$ . Therefore, $W_{8, 3}$ is not an even vertex magic. □

We are able to show that the necessary and sufficient condition for the $t$ -fold wheel $W_{n, t}$ is an even vertex magic for every even integer $4\leq n \leq 8$ .

Theorem 3.6. For every even integer $4\leq n \leq 8$ and integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic if and only if $t = n-2$ .

Proof. Let $n$ be an even integer where $4\leq n \leq 8$ and $t$ is an integer where $t \geq 2$ . Assume that the $t$ -fold wheel $W_{n, t}$ is an even vertex magic. By Proposition 2.2, $2\leq t \leq n$ .

Case 1. $n = 4, 6.$ If either $2\leq t \leq n-3$ or $n-1\leq t \leq n$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. Therefore, $t = n-2$ .

Case 2. $n = 8.$ If either $4\leq t \leq n-3$ or $n-1\leq t \leq n$ , then $n^2t^2+2n^2t+n$ is not divisible by $n+t$ , and hence $k$ is not an integer, which is a contradiction. If $t = 3$ , then, by Lemma 3.5, $W_{n, t}$ is not an even vertex magic, which is a contradiction. If $t = 2$ , then,

$2nt^3-n^2t^2-2n^2t+6nt^2+7nt+n^2-t^2+n-t-\frac{2nt^4+4nt^3-2nt}{n+t} = -62 < 0,$

which is a contradiction with Proposition 2.6. Therefore, $t = n-2$ .

Conversely, assume $t = n-2$ . By Theorem 3.4, $W_{n, t}$ is an even vertex magic. □

4. Conclusions

In this paper, we have not only established the bound of an integer $t$ for the even vertex magic total labeling of the $t$ -fold wheel, but have also presented the necessary condition for such labeling in terms of $n$ and $t$ . Furthermore, we have conducted an investigation into the sufficient conditions for labelings that can serve as even vertex magic total labelings for $W_{n, n}$ when $n$ is odd, and $W_{n, n-2}$ when $n$ is even.

Our research has led us to the following significant conclusions:

● For every integer $t \geq 2$ , the $t$ -fold wheel $W_{3, t}$ is an even vertex magic total labeling if and only if $t = 2, 3$ .

● For every odd integer $5 \leq n \leq 9$ and an integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic total labeling if and only if $t = n$ .

● For every even integer $4 \leq n \leq 8$ and an integer $t \geq 2$ , the $t$ -fold wheel $W_{n, t}$ is an even vertex magic total labeling if and only if $t = n-2$ .

In essence, our work has discussed the characterizations of $t$ -fold wheel $W_{n, t}$ to possess an even vertex magic total labeling for an integer $3 \leq n \leq 9$ . It would be interesting to apply the results of this paper to further study under what conditions for $W_{n, t}$ will be an even vertex magic, especially for a larger $n$ .

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no conflicts of interest.

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AIMS Mathematics

Browder spectra of closed upper triangular operator matrices

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1. Introduction

2. The conditions for an even vertex magic $W_{n, t}$

3. An even vertex magic $W_{n, t}$ where $3\leq n\leq 9$

4. Conclusions

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Browder spectra of closed upper triangular operator matrices

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2. The conditions for an even vertex magic $W_{n, t}$

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