Regular graphs with a complete bipartite graph as a star complement^††thanks: This work is supported by the National Natural Science Foundation of China (Grant No. 11971180), the Guangdong Provincial Natural Science Foundation (Grant No. 2019A1515012052).

Let $G$ be a simple graph with vertex set $V(G)=\left\{1,2,\dots,n\right\}=\left[n\right]$ and edge set $E(G)$. The adjacency matrix of $G$ is an $n\times n$ matrix $A(G)=(a_{ij})$, where $a_{ij}=1$ if vertex $i$ is adjacency to vertex $j$, and $0$ otherwise. We use the notation $i\sim j$ ($i\nsim j$) to indicate that $i,j$ are adjacent (not-adjacent) and the notation $d_{G}(i)$ to indicate the degree of vertex $i$ in $G$. The adjacency eigenvalues of $G$ are just the eigenvalues of $A(G)$. For more details on graph spectra, see [6].

Let $\mu$ be an eigenvalue of $G$ with multiplicity $k$. A star set for $\mu$ in $G$ is a subset $X$ of $V(G)$ such that $\left|X\right|=k$ and $\mu$ is not an eigenvalue of $G-X$, where $G-X$ is the subgraph of $G$ induced by $\overline{X}=V(G)\setminus X$. In this situation $H=G-X$ is called a star complement corresponding to $\mu$. Star sets and star complements exist for any eigenvalue of a graph, and they need not to be unique. The basic properties of star sets are established in Chapter $7$ of [7].

There is another equivalent geometric definition for star sets and star complements. Let $G$ be a graph with vertex set $V(G)=\left\{1,\dots,n\right\}$ and adjacency matrix $A$. Let $\left\{e_{1},\dots,e_{n}\right\}$ be the standard orthonormal basis of $\mathbb{R}^{n}$, $\mu$ be an eigenvalue of $G$, and $P$ be the matrix which represents the orthogonal projection of $\mathbb{R}^{n}$ onto the eigenspace $\mathcal{E}(\mu)=\left\{x\in\mathbb{R}^{n}:A(G)x=\mu x\ \right\}$ of $A$ with respect to $\left\{e_{1},\dots,e_{n}\right\}$. Since $\mathcal{E}(\mu)$ is spanned by the vectors $Pe_{j}(j=1,\dots,n)$, there exists $X\subseteq V(G)$ such that the vectors $Pe_{j}(j\in X)$ form a basis for $\mathcal{E}(\mu)$. Such a subset $X$ of $V(G)$ is called a star set for $\mu$ in $G$. In this situation $H=G-X$ is called a star complement for $\mu$.

For any graph $G$ of order $n$ with distinct eigenvalues $\lambda_{1},\dots,\lambda_{m}$, there exists a partition $V(G)=X_{1}\bigcup\cdots\bigcup X_{m}$ such that $X_{i}$ is a star set for eigenvalue $\lambda_{i}\ (i=1,\dots,m)$. Such a partition is called a star partition of $G$. For any graph $G$, there exists at least one star partition ([10]). Each star partition determines a basis for $\mathbb{R}^{n}$ consisting of eigenvectors of an adjacency matrix. It provides a strong link between graph structure and linear algebra.

There are a lot of literatures about using star complements to construct and characterize certain graphs([1, 2, 3, 8, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25]), especially, regular graphs with a prescribed graph such as $K_{1,s}$, $K_{1}\triangledown hK_{q}$, $K_{2,5}$, $K_{2,s}$, $K_{1,1,t}$, $K_{1,1,1,t}$, $\overline{sK_{1}\cup K_{t}}$, $P_{t}\ (\mu=1)$, $K_{r,r,r}\ (\mu=1)$ or $K_{r,s}+tK_{1}\ (\mu=1)$ as a star complement were well studied in the literature. Motivated by the above research, in this paper, we introduce the fundamental properties of the theory of star complements in Section 2, study the regular graphs with the bipartite graph $K_{t,s}\ (s\geq t\geq 1)$ as a star complement for an eigenvalue $\mu$ in Section 3, completely characterize the regular graphs with $K_{3,s}\ (s\geq 3)$ as a star complement for an eigenvalue $\mu$ in Section 4, study some properties of $K_{s,s}$ in Section 5, and propose some problems for further research.

In this section, we introduce some results of star sets and star complements that will be required in the sequel. The following fundamental result combines the Reconstruction Theorem ([7, Theorem 7.4.1]) with its converse ([7, Theorem 7.4.4]).

Note that if $X$ is a star set for $\mu$, then the corresponding star complement $H(=G-X)$ has adjacency matrix $C$, and (2.1) tells us that $G$ can be determined by $\mu$, $H$ and the $H$-neighbourhood of vertices in $X$, where the $H$-neighbourhood of the vertex $u\in X$, denoted by $N_{H}(u)$, is defined as $N_{H}(u)=\left\{v\mid v\sim u,v\in V(H)\right\}$.

It is usually convenient to apply (2.1) in the form

where $m(x)$ is the minimal polynomial of $C$. This is because $m(\mu)(\mu I-C)^{-1}$ is given explicitly as follows.

In order to find all the graphs with a prescribed star complement $H$ for $\mu$, we need to find all solution $A_{X}$, $B$ for given $\mu$ and $C$. For any $\textbf{x},\textbf{y}\in\mathbb{R}^{q}$, where $q=\left|V(H)\right|$, let

Let $\textbf{b}_{u}$ be the column of $B$ for any $u\in X$. By Theorem 2.1, we have

In view of the two equations in the above corollary, we have

It follows from $(2)$ of Lemma 2.4 that there are only finitely regular graphs with a prescribed star complement for $\mu\notin\left\{-1,0\right\}$. If $\mu=0$ and $X$ has distinct vertices $u$ and $v$ with the same neighbourhood in $G$, then $u$ and $v$ are called duplicate vertices. If $\mu=-1$ and $X$ has distinct vertices $u$ and $v$ with the same closed neighbourhood in $G$, then $u$ and $v$ are called co-duplicate vertices (see [11]).

Recall that if the eigenspace $\mathcal{E}(\mu)$ is orthogonal to the all-1 vector j then $\mu$ is called a non-main eigenvalue. From (2.2), we have the following result.

In the rest of this paper, we let $H\cong K_{t,s}\ (s\geq t\geq 1)$, $(V,W)$ be a bipartition of the graph $K_{t,s}$ with $V=\left\{v_{1},v_{2},\dots,v_{t}\right\}$, $W=\left\{w_{1},w_{2},\dots,w_{s}\right\}$. We say that a vertex $u\in X$ is of type $(a,b)$ if it has $a$ neighbours in $V$ and $b$ neighbours in $W$. Clearly $(a,b)\neq(0,0)$ and $0\leq a\leq t$, $0\leq b\leq s$.

Let $C$ be the adjacency matrix of $H$, then $C$ has minimal polynomial $m(x)=x(x^{2}-ts).$ Since $\mu$ is not an eigenvalue of $C$, we have $\mu\neq 0$ and $\mu^{2}\neq ts$. From Proposition 2.2, we have

If $\mu$ is a non-main eigenvalue of $G$, then by (2.4) and (2.5) we have

Using (2.5) to compute $\left\langle\textbf{b}_{u},\textbf{b}_{u}\right\rangle=\mu$, we obtain the following equation

Let $u,v$ be distinct vertices in $X$ of type $(a,b)$, $(c,d)$, respectively. Let $\rho_{uv}=\left|N_{H}(u)\cap N_{H}(v)\right|$, $a_{uv}=1$ or $0$ according as $u\sim v$ or $u\nsim v$. Using (2.5) to compute $\left\langle\textbf{b}_{u},\textbf{b}_{v}\right\rangle=-a_{uv}$, we have

An $r$-regular graph $G$ with $n$ vertices is said to be strongly regular with parameters $(n,r,e,f)$ if every two adjacent vertices in $G$ have $e$ common neighbours and every two non-adjacent vertices have $f$ common neighbours. For example, Petersen graph is strongly regular with parameters $(10,3,0,1)$.

For the regular graphs with the complete bipartite graph $K_{t,s}$ as a star complement, the case of $t=1$ was solved by Rowlinson and Tayfeh-Rezaie in 2010 ([19]), the case of $t=2,\ s=5$ was solved by Rowlinson and Jackson in 1999 ([18]), the case of $t=2,\ s\neq 5$ was solved by Yuan, Zhao, Liu and Chen in 2018 ([25]), and the conclusions are listed below.

In this section, we consider the general case. We prove that there is no regular graph $G$ with $K_{t,s}\ (s\geq t\geq 1)$ as a star complement for $\mu=-t$, characterize the graph $G$ when $\mu=r$, $\mu=-1$, and the case with all vertices in $X$ of type $(0,b)$ for $\mu\notin\{-t,r,-1\}$. Furthermore, we propose a question for further research.

If $t=1$, from Theorem 2.2 of [19], there is no $r$-regular graph $G$ with $K_{1,s}$ as a star complement for $\mu=-1$.

If $t\geq 2$, by Lemma 2.6, we know that $\mu=-t$ is a non-main eigenvalue of $G$, and by (2.6), we have

Since $s>t$ and $t\geq 2$, (3.1) implies that $a=t$, and thus $s(b-t^{2})=b^{2}-tb-t^{3}+t^{2}$ by (2.7).

If $b=t^{2}$, then $t^{4}-2t^{3}+t^{2}=0$, thus $t=0$ or $1$, a contradiction.

If $b\neq t^{2}$, then $s=\frac{b^{2}-tb-t^{3}+t^{2}}{b-t^{2}}=b-t^{2}+\frac{t^{2}(t-1)^{2}}{b-t^{2}}+2t^{2}-t$. If $b<t^{2}$, then $s\leq-2\sqrt{t^{2}(t-1)^{2}}+2t^{2}-t=t$, which contradicts with $s>t$. Thus $b>t^{2}$ and $s-(b+t-1)=\frac{b(t-1)^{2}}{b-t^{2}}>0$. Considering degrees, we have

and

Hence, $d_{G}(v_{1})=d_{G}(v_{2})=\cdots=d_{G}(v_{t})>d_{G}(u)$ which contradicts to the regularity of $G$.

Combining the above arguments, there is not an $r$-regular graph $G$ with $K_{t,s}\ (s\geq t\geq 1)$ as a star complement for $\mu=-t$.$\quad\square$

By $G$ is $r$-regular and connected, $\mu=r$, we know $k=1$ and then $\left|X\right|=1$. Since $G$ is regular, we have $d_{G}(v_{1})=d_{G}(v_{2})=\cdots=d_{G}(v_{t})$. Let $X=\left\{u\right\}$. Then either $u\sim v_{1}$, $u\sim v_{2}$, …,$u\sim v_{t}$, or $u\nsim v_{1}$, $u\nsim v_{2}$,…, $u\nsim v_{t}$.

If $u\sim v_{1}$, $u\sim v_{2}$, …,$u\sim v_{t}$, then $d_{G}(v_{1})=d_{G}(v_{2})=\cdots=d_{G}(v_{t})=s+1,$ which implies that $d_{G}(u)=s+1$. It follows that the vertex $u$ is adjacent to $s-t+1$($\geq 1$) vertices of $W$, and thus $t=1$ by $d_{G}(w_{1})=d_{G}(w_{2})=\cdots=d_{G}(w_{s})$. Since $d_{G}(w_{1})=t+1=d_{G}(v_{1})$, we have $s=t=1$ and $G\cong C_{3}$.

If $u\nsim v_{1}$, $u\nsim v_{2}$, …, $u\nsim v_{t}$, in view of the regularity, we have $d_{G}(u)=d_{G}(v_{1})=d_{G}(v_{2})=\cdots=d_{G}(v_{t})=s,$ and then $d_{G}(w_{1})=d_{G}(w_{2})=\cdots=d_{G}(w_{s})=t+1$. Hence we have $s=r=t+1$ and $G\cong K_{t+1,t+1}$.$\square$

Let $H\cong K_{t,s}$, $(V,W)$ be a bipartition of the graph $K_{t,s}$ with $V=\left\{v_{1},v_{2},\dots,v_{t}\right\}$, $W=\left\{w_{1},w_{2},\dots,w_{s}\right\}$. We obtain an $r$-regular graph $G(r)$ with $V(G(r))=X\cup V(H)$, $X=V_{1}\cup\cdots\cup V_{t}\cup W_{1}\cup\cdots\cup W_{s}$ where $V_{i}$ is the set of vertices of type $(1,s)$ adjacent to $v_{i}\in V$ with $|V_{i}|=(r+1)(s-1)/(ts-1)-1$, $V_{i}$ induces a clique for $1\leq i\leq t$, $W_{i}$ is the set of vertices of type $(t,1)$ adjacent to $w_{i}\in W$ with $|W_{i}|=(r+1)(t-1)/(ts-1)-1$, $W_{i}$ induces a clique for $1\leq i\leq s$. For any $i$, $j$, each vertex in $V_{i}$ is adjacent to all vertices in $W_{j}$.

The greatest common divisor of $a$ and $b$ is denoted by $\gcd(a,b)$. For $\mu=-1$, we have the following theorem.

By Lemma 2.6, we know that $\mu=-1$ is a non-main eigenvalue of $G$, thus from (2.6), we have

Combining (3.2) and (3.3), if $a=1$, then $b=s$; if $a=t$, then $b=1$; if $1<a<t$, then $b=a-t+1,\ s=2-t$ or $b=\frac{a}{t},\ s=\frac{1}{t}$. It is obvious that $s=2-t$ or $s=\frac{1}{t}$ contradicts with $s\in\mathbb{Z}_{+}$. Therefore, the possible types of vertices in $X$ are $(1,s)$, $(t,1)$, and the feasible solution of (2.8) are shown in Table 1.

We observe that when $u,v$ are of different types, they must be adjacent; when $u,v$ are of the same type, $u\sim v$ if and only if they have the same $H$-neighbourhoods, and thus $u,v$ are co-duplicate vertices. We can add arbitrarily many co-duplicate vertices when constructing graphs with a prescribed star complement for $-1$.

Now we partition the vertices in $X$. Let $V_{i}$ be the set of vertices of type $(1,s)$ in $X$ adjacent to $v_{i}\in V$, $W_{i}$ be the set of vertices of type $(t,1)$ in $X$ adjacent to $w_{i}\in W$. Then any two vertices in $V_{i}\ (W_{i})$ are co-duplicate vertices. We do not exclude the possibility that some of the sets $V_{i}$, $W_{i}$ are empty. Then for any $v_{i}\in V$, we have $d_{G}(v_{i})=s+|V_{i}|+\sum\limits_{i=1}^{s}|W_{i}|;$ and for any $w_{i}\in W$, we have $d_{G}(w_{i})=t+\sum\limits_{i=1}^{t}|V_{i}|+|W_{i}|.$

Since $G$ is $r$-regular, we have $|V_{1}|=|V_{2}|=\cdots=|V_{t}|$ by $d_{G}(v_{1})=d_{G}(v_{2})=\cdots=d_{G}(v_{t})$ and $|W_{1}|=|W_{2}|=\cdots=|W_{s}|$ by $d_{G}(w_{1})=d_{G}(w_{2})=\cdots=d_{G}(w_{s})$. Then we have

It turns out that

Since $|V_{1}|\in\mathbb{N}$, $|W_{1}|\in\mathbb{N}$ and

we have $r\equiv-1\ ({\rm mod}\ \frac{ts-1}{\gcd(s-1,t-1)})$. Consequently we obtain an $r$-regular graph $G(r)$.$\quad\square$

Next, we consider the case $\mu\notin\left\{-1,-t,r\right\}$. The following lemma lists all possible types of vertices in $X$.

For $\mu\notin\{-1,-t,r\}$, we consider the case $a=0$ in the following by Lemma 3.7.

Now we consider $H=K_{t,\mu\left(\mu^{2}+2t\mu+\mu+t^{2}\right)/t}$ and all vertices in $X$ are of type $(0,\mu^{2}+t\mu)$. Then $r=s$. Counting the edges between $X$ and $V(H)$, we have $\left|X\right|(\mu^{2}+t\mu)=s(r-t).$ Thus

Then $(\mu+t+1)\left(\mu^{3}+(2t+1)\mu^{2}+(t^{2}-t)\mu-t^{2}\right)=0$ from (3.4). When $\mu^{3}+(2t+1)\mu^{2}+(t^{2}-t)\mu-t^{2}=0$, then $\mu\notin\mathbb{Z}$ and thus $r=\frac{\mu}{t}\left(\mu^{2}+2t\mu+\mu+t^{2}\right)-\frac{1}{t}(\mu^{3}+(2t+1)\mu^{2}+(t^{2}-t)\mu-t^{2})=\mu+t\notin\mathbb{Z}$, a contradiction. When $\mu=-(t+1)$, then $r=s=t+1$ and $G\cong K_{t+1,t+1}$.