Group Constant-Sum Spectrum of Nearly Regular Graphs

Abstract

For an undirected graph G, a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum $\mathbb{A}$-flows, which gives edge weights using nonzero elements of an additive Abelian group $\mathbb{A}$ and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: $\mathbb{A}={\mathbb{Z}}_k$, the finite cyclic group of integer congruence modulo k, and $\mathbb{A}=\mathbb{Z}$, the infinite cyclic group of integers. The constant sum under a constant-sum $\mathbb{A}$-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by $I_k\left(G\right)$ and $I\left(G\right)$ for $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$, respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases $\mathbb{Z}$ and ${\mathbb{Z}}_k$ have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs $W_n$ and fan graphs $F_n$ in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning ${\mathbb{Z}}_k$ and $\mathbb{Z}$, respectively. Some open problems will be mentioned in the concluding remarks.

1. Introduction and Background

Throughout this article, only finite undirected graphs are considered. All graph theoretical terminology not defined here can be referred to in [1]. A graph G has the edge set $E\left(G\right)$ and vertex set $V\left(G\right)$. For any additive Abelian group $\mathbb{A}$, let ${\mathbb{A}}^{*}=\mathbb{A}-\left\{0\right\}$ be the set of all nonzero elements and 0 the additive identity of $\mathbb{A}$. Given a graph G, a mapping $f:E\left(G\right)\to {\mathbb{A}}^{*}$ is also called an edge labeling of G. A graph G admits a constant-sum $\mathbb{A}$-flow or is said to be $\mathbb{A}$-magic if there is an edge labeling such that the induced vertex labeling $f^{+}:V\left(G\right)\to \mathbb{A}$ defined by $f^{+}\left(v\right)={\sum }_{uv\in E\left(G\right)}f\left(uv\right)$ is a constant map. We call the constant a magic sum of G with respect to $\mathbb{A}$, or an index for short. We then denote the set of all constants r such that the graph G admits a constant-sum $\mathbb{A}$-flow with an index r by $I_{\mathbb{A}}\left(G\right)$ and call it the constant-sum spectrum, or the index spectrum, of G with respect to $\mathbb{A}$. Notice that, in this article, for convenience, we focus on two special cases: $\mathbb{A}={\mathbb{Z}}_k$, the finite cyclic group of integer congruence modulo k, and $\mathbb{A}=\mathbb{Z}$, the infinite cyclic group of integers. Generally, a graph may have more than one edge labeling to admit a constant-sum $\mathbb{A}$-flow or to be $\mathbb{A}$-magic. No generally efficient algorithm is known for finding such magic labeling for general graphs. A. Kotzig and A. Rosa used the same term in [2], but their notion of magic labeling is different from what we consider here as edge labeling. Instead, A. Kotzig and A. Rosa discussed the total labeling over vertices and edges. They say a graph has a magic labeling if for each edge, the sum of the edge label and its two end vertex labels is constant. It is well known [3,4,5] that a graph G is $\mathbb{N}$-magic if and only if every edge of G is contained in a $(1,2)$-factor and every pair of edges is separated by this $(1,2)$-factor, where $\mathbb{N}$ is the set of positive integers and a $(1,2)$-factor is a regular spanning subgraph of degree one or two for each of its components. For a list of properties of $\mathbb{N}$-magic graphs, see [6,7,8]. Richard Stanley first studied $\mathbb{Z}$-magic graphs in [9,10] and demonstrated that the study of magic labelings can be reduced to solving a system of linear Diophantine equations.

Note that one special case of magic sums of $\mathbb{A}$-magic graphs while $A=\mathbb{Z}$ with magic sum constant zero has been studied intensively, namely zero-sum flows, which was initially studied by S. Akbari et al. [11] in 2009. One may treat zero-sum flows as an undirected analog of nowhere-zero flows for directed graphs. A nowhere-zero flow in a directed graph is an assignment of nonzero integers to edges such that for every vertex, the sum of the incoming edge labels equals the sum of outgoing edge labels. A nowhere-zero k-flow is a nowhere-zero flow in which the assigned values are integers with absolute values less than k. A celebrated conjecture of Tutte in 1954 says that every bridgeless graph has a nowhere-zero 5-flow. There is a more general concept of a nowhere-zero flow that uses bidirected edges instead of directed ones, first systematically developed by Bouchet [12] in 1983. Bouchet raised the conjecture that every bidirected graph with a nowhere-zero integer flow has a nowhere-zero 6-flow, which is still unsettled. S. Akbari et al. raised a conjecture (called Zero-Sum 6-Flow Conjecture) in 2010 for zero-sum flows similar to Tutte’s 5-flow conjecture for nowhere-zero flows as follows: if G is a graph with a zero-sum flow, then G admits a zero-sum 6-flow. It was proved by Akbari et al. [13] that the above Zero-Sum 6-Flow Conjecture is equivalent to Bouchet’s 6-Flow Conjecture for bidirected graphs. The study of zero-sum flows was extended to constant-sum flows first by T.-M. Wang et al. [14] in 2011 and later studied by Akbari et al. [15,16,17,18] as 0-sum and 1-sum flows with related topics. Another extension of zero-sum flows to zero-sum flow numbers has also been initially studied by T.-M. and Wang et al. [19,20,21,22] over the years. More references can be seen in the survey article by Gallian [23].

The constant sum under a constant-sum $\mathbb{A}$-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant-sum spectrum and denoted by $I_{\mathbb{A}}\left(G\right)$. The constant-sum spectrum $I_{\mathbb{A}}\left(G\right)$ of various graph classes G was studied and calculated. We refer to $I_k\left(G\right)$ and $I\left(G\right)$ as the constant-sum spectrum with respect to ${\mathbb{Z}}_k$ and $\mathbb{Z}$, respectively, in this paper. Note that the case of constant-sum ${\mathbb{Z}}_2$-flows is not hard to calculate since every edge of a graph with ${\mathbb{Z}}_2$-flows must be labeled with 1, and hence, the constant-sum spectrum is completely determined by the degree sequence of the graph. Therefore, the constant-sum spectrum $I_k\left(G\right)$ is of interest for $k\ge 3$. In 2011 [14], T.-M. Wang et al. obtained the following result regarding the index sets of r-regular graphs (except the case $r=4k,k\ge 1$) of order n:

$$I\left(G\right)=\left\{\begin{array}{cc}{\mathbb{Z}}^{*}=\mathbb{Z}-\left\{\mathbf{0}\right\},\hfill & \hfill r=1.\\ \mathbb{Z},\hfill & \hfill r=2\mathrm{and}G\mathrm{contains}\mathrm{even}\mathrm{cycles}\mathrm{only}.\\ \mathbf{2}{\mathbb{Z}}^{*},\hfill & \hfill r=2\mathrm{and}G\mathrm{contains}\mathrm{an}\mathrm{odd}\mathrm{cycle}.\\ \mathbf{2}\mathbb{Z},\hfill & \hfill r\ge 3,r\mathrm{even}\mathrm{and}n\mathrm{odd}.\\ \mathbb{Z},\hfill & \hfill r\ge 3,r\ne 4k,\mathrm{and}n\mathrm{even}.\end{array}\right.$$

In 2023 [24], T.-M. Wang studied group constant-sum $\mathbb{A}$-flows while $\mathbb{A}={\mathbb{Z}}_k$, particularly for regular and nearly regular graphs. We initiated the study of basic properties for the constant-sum ${\mathbb{Z}}_k$-flows and extended to a more general situation constant-sum $\mathbb{A}$-flows for an Abelian group $\mathbb{A}$. Particularly in [24], we obtained that, for a regular graph G admitting a 1-factor, the constant-sum spectrum $I_k\left(G\right)$ is full ${\mathbb{Z}}_k$ for all $k\ge 3$. As a consequence, it is characterized for all 3-regular graphs with a full constant-sum spectrum. We also give examples of regular graphs without a 1-factor whose constant-sum spectrum is not full ${\mathbb{Z}}_k$ for some k. Among other results, we completely determine the constant-sum spectrum concerning ${\mathbb{Z}}_k$ of complete bipartite graphs $K_{m,n}$ for $m,n\ge 2$ as the additive cyclic subgroups of ${\mathbb{Z}}_k$ generated by ${\displaystyle \frac{k}{d}}$, where $d=gcd(m-n,k)$. Moreover, we also completely determine the magic sum spectrum of regular graphs of even degrees.

Inspired by the above studies for complete bipartite graphs $K_{m,n}$, which are nearly regular of possible two different degrees, two classes of nearly regular graphs such as wheels and fans admitting constant-sum $\mathbb{A}$-flows are studied for $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$, respectively. The fan graph $F_n$ is composed of a path $P_n$ on n vertices joined by another vertex, while the wheel graph $W_n$ is composed of a cycle $C_n$ on n vertices joined by another vertex. Among others, we completely determine the constant-sum spectrum of wheel graphs and fan graphs via a method of edge subdivision. In the concluding section, we mention more open problems that need to be explored further.

2. Basics for Constant-Sum $\mathbb{A}$-Flows

Recall that we have the following important necessary condition for a graph admitting constant-sum $\mathbb{A}$-flows with index r, which can be used to determine the magic sum spectrum.

Proposition 1. 

For any constant-sum $\mathbb{A}$-flows f of a graph $G=\left(V\right(G),E(G\left)\right)$ with index r, we have

$$2\sum _{e\in E\left(G\right)}f\left(e\right)=r·\left|V\left(G\right)\right|.$$

By the above necessary condition in general, for a graph G with $\left|V\right(G\left)\right|$ odd and for $\mathbb{A}=\mathbb{Z}$ and $\mathbb{A}={\mathbb{Z}}_k$ with even k, it implies that r must be an even number, as a representative in the congruence class modulo k. Therefore, we have the following general upper bounds for the constant-sum spectrum:

Corollary 1. 

For a graph G with odd vertices, we have that $I\left(G\right)⫅2\mathbb{Z}⫋\mathbb{Z}$.

Corollary 2. 

For a graph G with odd vertices and for even k, we have that $I_k\left(G\right)⫅2{\mathbb{Z}}_k⫋{\mathbb{Z}}_k$.

In order to obtain the whole constant-sum spectrum, we describe a subdivision method that is used in later sections for the construction of constant-sum $\mathbb{A}$-flow labeling with the same magic constant-sum index of an infinite family of graphs, in particular, fans and wheels. Let G be a graph with index r under a constant-sum $\mathbb{A}$-flow labeling f; using the subdivision method, we may obtain a new graph $G^{\prime }$ with a larger order, and a new constant-sum $\mathbb{A}$-flow $f^{\prime }$ on $G^{\prime }$ with the same index r, based upon G and f. We proceed by choosing in G a vertex v and edges $e_1,e_2$ with labels $f\left(e_2\right)=a$ and $f\left(e_1\right)=b$, which are not incident with v. Then, we subdivide these two edges by inserting new vertices of degree 2, joining them to v, respectively (see Figure 1). We may then construct a new labeling $f^{\prime }$ on $G^{\prime }$ by keeping the labels on G unchanged and labeling $r-2a$ and $r-2b$ on two newly inserted edges, respectively. Note that if $(r-2a)+(r-2b)\equiv 0$, $r-2a\ne 0$, and $r-2b\ne 0\left(modk\right)$, then the new labeling $f^{\prime }$ on $G^{\prime }$ is still constant-sum ${\mathbb{Z}}_k$-flow with index r, and $f^{\prime }\left(E\left(G^{\prime }\right)\right)=f\left(E\left(G\right)\right)\cup \{r-2a,r-2b\}$. Therefore, with this method, we can calculate the constant-sum spectrum of fans and wheels in the following sections.

Figure 1. General subdivision method.

3. Constant-Sum ${\mathbb{Z}}_{\mathit{k}}$-Flows of Fans ${\mathit{F}}_{\mathit{n}}$

We are in a position to calculate the constant-sum spectrum of fans $F_n$ in this section and start with a lemma for ${\mathbb{Z}}_3$-magicness with zero sums. The constant-sum spectrum of 3-fan $F_3$ is special and different from other cases, as shown in the following.

Lemma 1. 

For the 3-fan $F_3$, we have the following facts:

1. 

$F_3$ is constant-sum ${\mathbb{Z}}_k$-flow if and only if k is even and $k\ge 2$. Therefore, $I_k\left(F_3\right)=\varnothing$ for k odd.

2. 

$I_4\left(F_3\right)=\{0,2\}$.

3. 

For $k\ge 5$, $I_k\left(F_3\right)={\mathbb{Z}}_k$, for k even.

Proof. 

In $F_3$, as in Figure 2, we have that if r is an index, then there exists nonzero elements x and y in ${\mathbb{Z}}_k$ such that $2(x+y-r)=0$ and $x+y-r\ne 0$, $r-x\ne 0$, $r-y\ne 0$ $\left(modk\right)$; hence, k is even since $x+y-r$ is an element of order 2. Therefore, $F_3$ is constant-sum ${\mathbb{Z}}_k$-flow if and only if k is even and $k\ge 2$.

Figure 2. Constant-sum r of $F_3$.

Suppose that $r=1\in I_4\left(F_3\right)$, we have the labels on the path as $1-x\ne 0,1-y\ne 0$, and the labels on the spokes are $x\ne 0,y\ne 0,1-x-y=x+y-1\ne 0$. Then, we have $2(x+y)=2$; hence, $x+y=-1$, which is a contradiction to $x\ne 0,1$ and $y\ne 0,1$. Therefore, $\pm 1\notin I_4\left(F_3\right)$.

For an even $k\ge 5$, in order to realize all indices in $F_3$, we take $x=1,y=r-1+{\displaystyle \frac{k}{2}}$ for the index $r\ne 1$, and $x=2,y=r-2+{\displaystyle \frac{k}{2}}$, for $r=1$. □

Lemma 2. 

For $n\ge 3$, $0\in I_3\left(F_n\right)$, if and only if $n\equiv 1\left(mod3\right)$.

Proof. 

Assume that $0\in I_3\left(F_n\right)$. First, ensure that the edge labels on the path of the $F_n$ are all the same, either all 1’s or all $(-1)$’s. If not, then there will be 0-edges over the spokes. Without loss of generality, suppose the edge labels on the path are all 1’s. Thus, the labels on the spokes are two $(-1)$’s, and $n-2$ of them are 1’s. Then, by calculating the vertex sum of the center, we have $(n-2)-2\equiv n-1\equiv 0\left(mod3\right)$. Conversely, it follows from the given labeling. □

Lemma 3. 

$0\in I_k\left(F_n\right)$, for $n\ge 4$ and $k\ge 4$.

Proof. 

Please see Figure 3, where we have the labeling for $F_4,F_5,F_6$ such that the vertex sum is 0. Using the above subdivision method, we construct constant-sum ${\mathbb{Z}}_k$-flow labeling from $F_n$ to $F_{n+2}$ by subdividing one pair of 1-edge and $(-1)$-edge in $F_5$ and $F_6$, respectively. □

Figure 3. ${\mathbb{Z}}_k$-magic sum 0 of $F_4,F_5,F_6$, $k\ge 4$.

For other cases of fans $F_n$, see the following. We split the discussion into cases ${\mathbb{Z}}_3$-magic, ${\mathbb{Z}}_4$-magic, and ${\mathbb{Z}}_k$-magic for $k\ge 5$.

Lemma 4. 

$\pm 1\in I_3\left(F_n\right)$ if and only if $n\ge 6$.

Proof. 

Without loss of generality, it suffices to consider the case of index 1. The labeling on the edges incident to vertices of degree 2 must be both $-1$. One may easily check that $\pm 1\notin I_3\left(F_n\right)$ for $n<6$, and we have ${\mathbb{Z}}_3$-magic labeling with an index 1 of $F_6,F_7,F_8$ as in Figure 4.

Figure 4. ${\mathbb{Z}}_3$-magic sum 1 of $F_6,F_7,F_8$, and construction from $F_n$ to $F_{n+3}$ for $n\ge 6$.

To obtain ${\mathbb{Z}}_3$-magic labeling with index 1 from $F_n$ to $F_{n+3}$ for $n\ge 6$, we insert three vertices of degree two on some 1-edge on the path of $F_n$, join them to the center, and label $-1$ on the newly added spokes, as in Figure 4. Then, the resulting labeling will perform the task. □

Note that $F_n$ has an $n+1$ vertex, and thus, by the necessary condition Proposition 1, $\pm 1\notin I_4\left(F_n\right)$ for n even. Therefore, we consider the following ${\mathbb{Z}}_4$-magicness of $F_n$.

Lemma 5. 

$\pm 1\in I_4\left(F_n\right)$ for $n\ge 5$ is odd.

Proof. 

For $n\ge 5$ is odd, we have $1\in I_4\left(F_5\right)$ as the labeling given in Figure 5.

Figure 5. ${\mathbb{Z}}_4$-magic with index $\pm 1$ for $F_n$ for $n\ge 5$ odd.

Since there are edges labeled by 1 and 2 over the path of $F_n$, respectively, the subdivision method mentioned above may be used by subdividing the 1-edge and 2-edge and connect the newly inserted degree 2 vertices to the center. Then, we may construct ${\mathbb{Z}}_4$-magic labeling with index 1, and hence, $-1$ for $n\ge 5$ odd. □

Lemma 6. 

$2\in I_4\left(F_n\right)$ for $n\ge 3$.

Proof. 

As in Figure 6, first, we label $1,-1,1,-1\cdots$ alternatively on the path of $F_n$. For n odd, in order to obtain the index 2, the labeling on spokes has to be one of 1, $(n-2)$ of 2, and one of $-1$. For n even, the labeling on spokes has to be two of 1 and $(n-2)$ of 2. In both cases, the vertex sums are constant 2 and we have completed our calculation.

□

Figure 6. ${\mathbb{Z}}_4$-magic with index 2 for $F_n$.

In the following, we deal with the constant-sum ${\mathbb{Z}}_k$-flow case for $k\ge 5$. Again, by the necessary condition of being ${\mathbb{Z}}_k$-magic with index r in Proposition 1, we have that $I_k\left(F_n\right)\subset 2{\mathbb{Z}}_k$, for n even. We will realize every nonzero index $r=2a$, where $a\in {\mathbb{Z}}_k^{*}$ in the following lemma.

Lemma 7. 

For all $a\in {\mathbb{Z}}_k^{*}$, we have $2a\in I_k\left(F_n\right)$, for all $n\ge 4$, $k\ge 5$, and n even.

Proof. 

Let $n=2t$ and $F_n=\left\{v\right\}+\{u_1{u}_2\cdots u_{2t}\}$; we first delete the edges $u_{2i}{u}_{2i+1}$, $i=1,2,\cdots ,t-1$ alternatively, and obtain a windmill M, which is Eulerian since every vertex is of even degree, as in Figure 7.

Figure 7. Constant-sum spectrum of fans $F_n$, n even.

• Case 1: M contains an even number of triangles.

Hence, M is of even size, and we may label $\pm 1$ on one Eulerian tour of M such that vertex sum is 0 on each vertex. Add extra a to the $\pm 1$ labeling on the edges $v{u}_1$, $v{u}_{2t}$, and $u_{2j-1}{u}_{2j}$, $j=1,2,\cdots ,t$; then, we have labels $a\pm 1$ on M such that vertex sums on v, $u_1$, and $u_{2t}$ are $2a$, and are a on other vertices (see Figure 7). Now, return the edges $u_{2i}{u}_{2i+1}$, $i=1,2,\cdots ,t-1$ and label a on each of them; then, we have a labeling called $L_1$ on $F_n$ with $a,a\pm 1$ such that each vertex sum is $2a$.

By a similar method as above, except labeling $\pm 2$ instead on the Eulerian tour of M, we then have another labeling $L_2$ on $F_n$ with $a,a\pm 2$ such that each vertex sum is $2a$. Note that for $a\ne 0$ and $k>3$, if $a\pm 1=0$ in $L_1$, then we use the nonzero ${\mathbb{Z}}_k$-labeling $L_2$. If $a\pm 2=0$ in $L_2$, we use the nonzero ${\mathbb{Z}}_k$-labeling $L_1$. Thus, we have finished in this case.

• Case 2: M contains odd numbers (at least three) of triangles.

Take three triangles containing vertices $u_1$ and $u_{2t}$ of M, and label them as in Figure 7 such that the partial vertex sums on v, $u_1$, and $u_{2t}$ are 2, respectively, and 0 on other vertices. Note that the other triangles form an Euler graph of even size, and we label $\pm 1$ on them such that the partial vertex sum is 0 for each vertex on one Eulerian tour, and remains 2 on the center. Put the edges $u_{2i}{u}_{2i+1},i=1,2,\cdots ,t-1$ back and label 2 on them; we then have labels $2,\pm 1$ on $F_n$ such that each vertex sum is 2. By multiplying all labels on each edge by a, we may have labels $2a,\pm a$ such that each vertex sum is $2a$. Then, we have finished in this case. □

Lemma 8. 

$I_k\left(F_n\right)={\mathbb{Z}}_k$, for all $n\ge 5$ odd, and $k\ge 5$.

Proof. 

At first, we construct four types of labeling $L_1$, $L_2$, $L_3$, and $L_4$ such that vertex sum is r for $F_5$, as in Figure 8.

Figure 8. $L_1$, $L_2$, $L_3$, and $L_4$.

We use the subdivision method in the following to extend $L_1$, $L_2$, $L_3$, and $L_4$ to the labeling of $F_n$, for $n\ge 5$ and n odd. Note that via subdividing edges labeled by $-1$ and $r+1$ on $L_1$, $L_2$, respectively, we have the following sets of possible labels $S_1=\{r\pm 1,r-2,\pm (r+2)\}$ and $S_2=\{r+1,r-3,\pm (r+2),r-2\}$, respectively. Subdividing edges labeled by 1 and $r-1$ on $L_3$ and $L_4$, respectively, we have the following sets of possible labels $S_3=\{r-1,r+3,\pm (r-2\},r+2\}$ and $S_4=\{r,r\pm 1,\pm (r-2)\}$, respectively. Subdividing edges labeled by $-1$ and $r+1$ in $L_4$ again, we then have the sets of possible labels $S_5=\{r,r\pm 1,\pm (r+2)\}$.

It remains to show that the above sets of possible labels contain no common zero elements in ${\mathbb{Z}}_k$. It is not hard to see that $S_1$ contains zero elements only for $r=\pm 1,\pm 2$, and then, instead, we may use the following labeling: $S_2=\{-1,2,-2,\pm 3\}$ for $r=1$, $S_3=\{1,-2,2,\pm 3\}$ for $r=-1$, $S_4=\{-1,-3,-2,\pm 4\}$ for $r=-2$, and $S_5=\{2,1,3,\pm 4\}$ for $r=2$. Hence, we have completed our calculations. □

To summarize from the above lemmas, we have the following.

Theorem 1. 

The constant-sum spectrum of fans $F_n$, $n\ge 3$, for ${\mathbb{Z}}_k$ are as follows:

When $k=3,$

$$I_3\left(F_n\right)=\left\{\begin{array}{c}\varnothing ,n=3.\\ \left\{0\right\},n=4.\\ \varnothing ,n=5.\\ {\mathbb{Z}}_3,for all n\ge 6 and n\equiv 1\left(mod3\right).\\ {\mathbb{Z}}_3\setminus \left\{0\right\},for all n\ge 6 and n\equiv 0,2\left(mod3\right).\end{array}\right.$$

When $k=4$,

$$I_4\left(F_n\right)=\left\{\begin{array}{c}\{0,2\},n=3.\\ 2{\mathbb{Z}}_4=\{0,2\},n even.\\ {\mathbb{Z}}_4,n\ge 5 odd.\end{array}\right.$$

When $k\ge 5$,

$$I_k\left(F_n\right)=\left\{\begin{array}{c}{\mathbb{Z}}_k,for all n\ge 5 odd, and k\ge 5.\\ 2{\mathbb{Z}}_k,for all n\ge 4 even, and k\ge 5.\end{array}\right.$$

Note that $2{\mathbb{Z}}_k={\mathbb{Z}}_k$ when k is odd, and $2{\mathbb{Z}}_k=\{0,2,\cdots ,{\displaystyle \frac{k}{2}}\}$ when k is even.

4. Constant-Sum Spectrum of Wheels ${\mathit{W}}_{\mathit{n}}$ for $\mathbb{A}={\mathbb{Z}}_{\mathit{k}}$

To obtain the constant-sum spectrum of the wheel graphs $W_n$, $n\ge 3$, we look at the magic sum index 0. Note that the magic zero-sum was calculated in [25], and we include the proofs here for completeness. We consider the ${\mathbb{Z}}_3$-magic case first.

Lemma 9. 

For $n\ge 3$, $0\in I_3\left(W_n\right)$ if and only if $n\equiv 0$ (mod 3).

Proof. 

Assume that $0\in I_3\left(W_n\right)$. First, we observe that the edge labels on the cycle of the $W_n$ have to all be the same, either all 1’s or all $(-1)$’s. If not, then there will be 0-edges over the spokes. By symmetry, suppose the edge labels on the cycle are all 1’s. Thus, the labels on the spokes are all $(-1)$’s. Then, by calculating the vertex sum of the center, we have $(-1)·n\equiv 0\left(mod3\right)$, and hence, $n\equiv 0\left(mod3\right)$. The converse follows from the given labeling. □

In the following, we calculate the constant-sum spectrum of the wheels for the remaining cases $k\ge 4$.

Lemma 10. 

Let $n\ge 3$. Then $0\in I_k\left(W_n\right)$ for all $k\ge 4$.

Proof. 

We deal with the problem by using induction on $n\ge 6$ from n to $n+2$, and for the cases $W_n$ for $n\le 5$, we give the labeling in the figures. Note that $W_3$ is ${\mathbb{Z}}_k$-magic with zero sum for $k\ge 3$ (see Figure 9). In the case of $W_4$, by Lemma 9 and see Figure 9, we see that $0\in I_k\left(W_4\right)$ for all $k\ge 4$. For the case $W_5$, see Figure 9. The case on the left of Figure 9 is for ${\mathbb{Z}}_k$-magicness with zero-sum for $W_5$, for all $k\ge 5$, and the case on the right of Figure 9 is for ${\mathbb{Z}}_4$-magicness with zero sum for $W_5$.

Figure 9. $W_3$, $W_4$, and $W_5$.

The cases for $W_6$ and $W_7$ are presented in Figure 10. Combined with the Lemma 9, we see that $0\in I_k\left(W_6\right)$ for all $k\ge 3$ and $0\in I_k\left(W_7\right)$ for all $k\ge 4$. Assume the result is true for $n\ge 6$. The induction step is the construction from $W_n$ to $W_{n+2}$, for all $n\ge 6$, using the subdivision method mentioned above. Note that for the labeling given in Figure 10, $W_6$ and $W_7$ both have 1 and $-1$ over the cycles of wheels, respectively. Therefore, we may complete the constant-sum spectrum of $W_n$, for all $n\ge 6$, by subdividing the 1-edge and $(-1)$-edge. □

Figure 10. $W_6$ and $W_7$.

Lemma 11. 

$\pm 1\in I_3\left(W_n\right)$ for $n\ge 3$.

Proof. 

Suppose that $n+1\equiv r$ mod 3, and $r=0,1$, or 2. Label $-1$ on r edges of the outer cycle of $W_n$ non-consecutively, and 1 on other edges. Please see Figure 11. To obtain index 1, label $2r$ of 1’s and $n-2r$ of $(-1)$’s on the spokes, and hence, the vertex sum of the center is $2r-(n-2r)\equiv 4r-n\equiv 1\left(mod3\right)$. □

Figure 11. ${\mathbb{Z}}_3$-magic sum of 1 of $W_n$.

Lemma 12. 

For n odd, $I_k\left(W_n\right)={\mathbb{Z}}_k$.

Proof. 

Let $n=2t+1$. In $W_{2t+1}$, each vertex is of odd degree, and it admits a perfect matching P such that $W_{2t+1}\setminus P$ is an Euler graph of even size. Label $\pm 1$ on $W_{2t+1}\setminus P$ such that each partial vertex sum is 0, and then label the nonzero element r on the matching P; we thus obtain all possible labeling of index $r\ne 0$. The case for zero index is thus complete. □

By Proposition 1, we have $I_k\left(W_n\right)\subseteq 2{\mathbb{Z}}_k$ for n even. The following Lemma shows the constant-sum spectrum is the whole $2{\mathbb{Z}}_k$.

Lemma 13. 

For n even, $I_k\left(W_n\right)=2{\mathbb{Z}}_k$.

Proof. 

Let $n=2t$. Since the zero index case is completed, let $2a\ne 0$ in $2{\mathbb{Z}}_k$. Let $W_n=\left\{v\right\}+\{u_1{u}_2\cdots u_{2t-1}{u}_{2t}{u}_1\}$; we delete the edges $u_{2i}{u}_{2i+1},i=1,2,\cdots ,t-1$ and $u_{2t}{u}_1$ alternatively, and obtain a windmill M (see Figure 12).

Figure 12. Constant-sum spectrum of wheels $W_n$, n even.

• Case 1: M contains an even number of triangles.

Since M is of even size, we may label $\pm 1$ on one Eulerian tour of M such that the partial vertex sum is 0 on each vertex. Add value a to labels on edges $v{u}_1,v{u}_2,u_{2j-1}{u}_{2j}$, $j=2,\cdots ,t$, then we have labels $a\pm 1$ and $\pm 1$ on M such that partial vertex sums are $2a$ on v, and a on other vertices. Now, put back the edges $u_{2i}{u}_{2i+1},i=1,2,\cdots ,t-1$ and $u_{2t}{u}_1$, and then label a on them; we have labels $a,a\pm 1$ on $W_n$ such that each vertex sum is $2a$.

To remedy the case when $a\pm 1=0$, similarly, we put labels $\pm 2$ on the Eulerian tour of M, and we then have labels $a,a\pm 2$ and $\pm 2$ on $W_n$ such that each vertex sum is $2a$. Note that at least one of the two labeling on $W_n$ contains no zero element since $a\ne 0$ and $k\ge 3$ are under consideration.

• Case 2: M contains an odd number of triangles.

We have an Eulerian tour of odd size starting and ending at v and the edges are consecutively labeled $1,-1,\cdots ,1,-1,1$. Then, we have label $\pm 1$ on M such that the partial vertex sum is 2 on v and is 0 on other vertices. Put the $u_{2i}{u}_{2i+1},i=1,2,\cdots ,t-1$ and $u_{2t}{u}_1$ back as in previous cases and label 2 on them; we then have the labels $2,\pm 1$ on $W_n$ such that each vertex sum is 2. Now adjust the labeling by multiplying the labels on each edge by a; we may have the labeling with $2a,\pm a$ such that each vertex sum is $2a$. We are thus finished in this case. □

To summarize the above Lemmas, we have the following:

Theorem 2. 

The constant-sum spectrum of wheels $W_n$, $n\ge 3$, for ${\mathbb{Z}}_k$ are as follows:

when $k=3$,

$$I_3\left(W_n\right)=\left\{\begin{array}{c}{\mathbb{Z}}_3,n\equiv 0\left(mod3\right).\\ {\mathbb{Z}}_3\setminus \left\{0\right\},n\equiv 1,2\left(mod3\right).\end{array}\right.$$

when $k\ge 4$,

$$I_k\left(W_n\right)=\left\{\begin{array}{c}{\mathbb{Z}}_k,for all n even.\\ 2{\mathbb{Z}}_k,for all n odd.\end{array}\right.$$

Again, note that $2{\mathbb{Z}}_k={\mathbb{Z}}_k$ when k is odd, and $2{\mathbb{Z}}_k=\{0,2,\cdots ,{\displaystyle \frac{k}{2}}\}$ when k is even.

5. Constant-Sum $\mathbb{A}$-Flows of Fans and Wheels for $\mathbb{A}=\mathbb{Z}$

In order to obtain the constant-sum spectrum of fans and wheels for the case $\mathbb{A}=\mathbb{Z}$, we use the mathematical induction with a subdivision method by inserting new vertices and new edges to perform the task. We start with the fan graphs in the following.

5.1. Constant-Sum $\mathbb{A}$-Flows of Fans for $\mathbb{A}=\mathbb{Z}$

We calculate the constant-sum spectrum of fans $F_n$, for $n\ge 3$ in the following.

Lemma 14. 

The constant-sum spectrum of $F_3$ is empty.

Proof. 

Suppose $F_3$ has some index $r\in \mathbb{Z}$, as in Figure 13. Note that one has $x+y+z=r$ and $r-x+z+r-y=r$, which implies $z=0$, a contradiction. □

Figure 13. $F_3$ and the labels.

Lemma 15. 

$I\left(F_{2n+1}\right)=\mathbb{Z}$ for all $n\ge 2$.

Proof. 

We use induction from $F_{2n+1}$ to $F_{2n+3}$ for $n\ge 2$. Figure 14 shows the edge insertion from $F_5$ to $F_7$, which is also valid for general cases by induction. Therefore, $I\left(F_{2n+1}\right)=\mathbb{Z}$ for all $n\ge 2$. □

Figure 14. 0-sum 4-flow and 1-sum 5-flow from $F_{2n+1}$ to $F_{2n+3}$.

Note that since $\left|V\right(F_{2n}\left)\right|$ is odd, by Corollary 1, the index of $F_{2n}$ must be even. Therefore, for $F_{2n}$, we need to only consider even indices.

Lemma 16. 

$I\left(F_{2n}\right)=2\mathbb{Z}$ for all $n\ge 2$.

Proof. 

We use induction from $F_{2n}$ to $F_{2n+2}$ for $n\ge 4$. In the following, Figure 15 shows the edge insertion from $F_6$ to $F_8$, which is also valid for general cases.

Figure 15. 0-sum 4-flow and 2-sum 5-flow from $F_{2n}$ to $F_{2n+2}$.

By this, we show that $I\left(F_{2n}\right)=\mathbb{Z}$, for all $n\ge 3$. The only remaining case is $F_4$. See the instances of 0-sum 3-flow and 2-sum 4-flow for $F_4$, respectively, in Figure 16.

Figure 16. 0-sum 3-flow and 2-sum 4-flow for $F_4$.

Therefore, we completed our calculations. □

5.2. Constant-Sum $\mathbb{A}$-Flows of Wheels for $\mathbb{A}=\mathbb{Z}$

We calculate the constant-sum spectrum of wheels $W_n,n\ge 3$ for $\mathbb{A}=\mathbb{Z}$ in the following.

Lemma 17. 

$I\left(W_{2n+1}\right)=\mathbb{Z}$ for all $n\ge 1$.

Proof. 

We use the edge insertion method from $W_n$ to $W_{n+2}$ inductively for 0-sums and 1-sums (as shown in Figure 17 and Figure 18, respectively) as follows.

Figure 17. Wheels for index 0.

Figure 18. Wheels for index 1.

Note that in case $n=1$, $W_3$ is 3-regular and was calculated in [24]. Instead, we start the induction from $n=2$. See Figure 19 for instances of 0-sum 5-flow and 1-sum 4-flow for $W_5$, respectively. Therefore, $I\left(W_{2n+1}\right)=\mathbb{Z}$ for all $n\ge 1$. □

Figure 19. 0-sum 5-flow and 1-sum 4-flow for $W_5$.

Since $\left|V\right(W_{2n}\left)\right|$ is odd, by Corollary 1, the indices of $W_{2n}$ must be even. We then have the following Lemma:

Lemma 18. 

$I\left(W_{2n}\right)=2\mathbb{Z}$ for all $n\ge 2$.

Proof. 

Similarly, we use the edge insertion method from $W_n$ to $W_{n+2}$ inductively for 0-sums and 2-sums (as shown in Figure 17 and Figure 20, respectively) as follows.

Figure 20. Wheels for index 2.

We start the induction from $n=2$, which is $W_4$ in Figure 21.

Figure 21. 0-sum 4-flow and 2-sum 7-flow for $W_4$.

Therefore, $I\left(W_{2n}\right)=\mathbb{Z}$ for all odd $n\ge 2$. □

To summarize from the above Lemmas, we have obtained all constant-sum spectra of fans $F_n$ and wheels $W_n$ as follows.

Theorem 3. 

The constant-sum spectrum of the fan graphs $F_n$ is as follows:

$$I\left(F_n\right)=\left\{\begin{array}{c}\varnothing ,n=3.\\ 2\mathbb{Z},n=2k,k\ge 2.\\ \mathbb{Z},n=2k+1,k\ge 2.\end{array}\right.$$

Theorem 4. 

The constant-sum spectrum of wheel graphs $W_n$ is as follows:

$$I\left(W_n\right)=\left\{\begin{array}{c}2\mathbb{Z},n=2k,k\ge 2.\\ \mathbb{Z},n=2k+1,k\ge 1.\end{array}\right.$$

6. Concluding Remarks and Future Work

In this article, we completely determine the constant-sum spectrum of two nearly regular graph classes, namely fans and wheels, for $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$, respectively. We list the following open problems along the directions of recent research to be explored further, inspired by our work:

(1)

To calculate the constant-sum spectra of regular graphs of odd degrees and the constant-sum spectra of nearly regular graph classes in general.

(2)

To determine the constant-sum spectra of various general graph products (e.g., Cartesian product, lexicographic product, strong product, and corona product) of (regular) graphs and other graphs that contain special cases of regular graphs or nearly regular graphs such as complete bipartite graphs $K_{m,n}$, fan graphs $F_n$, wheel graphs $W_n$, etc.

(3)

Other than the complete bipartite graphs $K_{m,n}$, to identify other classes of graphs whose constant-sum spectrum are the cyclic subgroups of ${\mathbb{Z}}_k$. More generally identify classes of graphs whose constant-sum spectrum for an Abelian group $\mathbb{A}$ as the subgroup of $\mathbb{A}$.

(4)

Extending the work in this paper to the general case of Abelian group $\mathbb{A}$ based upon the special cases $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$.

(5)

For zero-sum flows, one has a more general parameter zero-sum flow number of a graph G, which is defined by the minimum k for G admitting a zero-sum k-flow. Similarly, one may study the parameter 1-sum flow number of a graph G, which is defined by the minimum k for G admitting a 1-sum k-flow.