Ordering the oriented unicyclic graphs whose skew-spectral radius is bounded by 2

Abstract

Let $S(G^{\sigma })$ be the skew-adjacency matrix of an oriented graph $G^{\sigma }$ with n vertices, and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of $S(G^{\sigma })$. The skew-spectral radius ${\rho }_s(G^{\sigma })$ of $G^{\sigma }$ is defined as $max\{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. A connected graph, in which the number of edges equals the number of vertices, is called a unicyclic graph. In this paper, the structure of oriented unicyclic graphs whose skew-spectral radius does not exceed 2 is investigated. We order all the oriented unicyclic graphs with n vertices whose skew-spectral radius is bounded by 2.

MSC:05C50, 15A18.

1 Introduction

Let G be a simple graph with n vertices. The adjacency matrix $A=A(G)$ is the symmetric matrix ${[a_{ij}]}_{n\times n}$, where $a_{ij}=a_{ji}=1$ if $v_i{v}_j$ is an edge of G, otherwise, $a_{ij}=a_{ji}=0$. We call $det(\lambda I-A)$ the characteristic polynomial of G, denoted by $\varphi (G;\lambda )$ (or abbreviated to $\varphi (G)$). Since A is symmetric, its eigenvalues ${\lambda }_1(G),{\lambda }_2(G),\dots ,{\lambda }_n(G)$ are real, and we assume that ${\lambda }_1(G)\ge {\lambda }_2(G)\ge \cdots \ge {\lambda }_n(G)$. We call $\rho (G)={\lambda }_1(G)$ the adjacency spectral radius of G.

The class of all graphs G whose largest (adjacency) spectral radius is bounded by 2 has been completely determined by Smith; see, for example, [1, 2]. Later, Hoffman [3], Cvetković et al. [4] gave a nearly complete description of all graphs G with $2<\rho (G)<\sqrt{2+\sqrt{5}}$ (≈2.0582). Their description was completed by Brouwer and Neumaier [5]. Then Woo and Neumaier [6] investigated the structure of graphs G with $\sqrt{2+\sqrt{5}}<{\lambda }_{\max }(G)<\frac{3}{2}\sqrt{2}$ (≈2.1312), Wang et al. [7] investigated the structure of graphs whose largest eigenvalue is close to $\frac{3}{2}\sqrt{2}$.

Another interesting problem that arises in the context of graph eigenvalues is to order graphs in some class with respect to the spectral radius or least eigenvalue. In 2003, Guo [8] gave the first six unicyclic graphs of order n with larger spectral radius. Belardo et al. [9] ordered graphs with spectral radius in the interval $(2,\sqrt{2+\sqrt{5}})$. In the paper [10], the first five unicyclic graphs on order n in terms of their smaller least eigenvalues were determined.

The graph obtained from a simple undirected graph by assigning an orientation to each of its edges is referred as the oriented graph. Let $G^{\sigma }$ be an oriented graph with vertex set $\{v_1,v_2,\dots ,v_n\}$ and edge set $E(G^{\sigma })$. The skew-adjacency matrix $S=S(G^{\sigma })={[s_{ij}]}_{n\times n}$ related to $G^{\sigma }$ is defined as

where (note that the definition is slightly different from the one of the normal skew-adjacency matrix given by Adiga et al. [11]). Since $S(G^{\sigma })$ is an Hermitian matrix, the eigenvalues ${\lambda }_1(G^{\sigma }),{\lambda }_2(G^{\sigma }),\dots ,{\lambda }_n(G^{\sigma })$ of $S(G^{\sigma })$ are all real numbers and, thus, can be arranged non-increase as

$${\lambda }_1\left(G^{\sigma }\right)\ge {\lambda }_2\left(G^{\sigma }\right)\ge \cdots \ge {\lambda }_n\left(G^{\sigma }\right).$$

The skew-spectral radius and the skew-characteristic polynomial of $G^{\sigma }$ are defined respectively as

$${\rho }_s\left(G^{\sigma }\right)=max\{\left|{\lambda }_1\left(G^{\sigma }\right)\right|,\left|{\lambda }_2\left(G^{\sigma }\right)\right|,\dots ,\left|{\lambda }_n\left(G^{\sigma }\right)\right|\}$$

and

$$\varphi (G^{\sigma };\lambda )=det(\lambda I_n-S\left(G^{\sigma }\right)).$$

Recently, much attention has been devoted to the skew-adjacency matrix of an oriented graph. In 2009, Shader and So [12] investigated the spectra of the skew-adjacency matrix of an oriented graph. In 2010, Adiga et al. [11] discussed the properties of the skew-energy of an oriented graph. In papers [13, 14], all the coefficients of the skew-characteristic polynomial of $G^{\sigma }$ in terms of G were interpreted. Cavers et al. [15] discussed the graphs whose skew-adjacency matrices are all cospectral, and the relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew-adjacency matrices. In [16], the author established a relation between ${\rho }_s(G^{\sigma })$ and $\rho (G)$. Also, the author gave some results on the skew-spectral radii of $G^{\sigma }$ and its oriented subgraphs.

A connected graph in which the number of edges equals the number of vertices is called a unicyclic graph. In this paper, we will investigate the skew-spectral radius of an oriented unicyclic graph. The rest of this paper is organized as follows: In Section 2, we introduce some notations and preliminary results. In Section 3, all the oriented unicyclic graphs whose skew-spectral radius does not exceed 2 are determined. The result tells us that there is a big difference between the (adjacency) spectral radius of an undirected graph and the skew-spectral radius of its corresponding oriented graph. Furthermore, we order all the oriented unicyclic graphs with n vertices whose skew-spectral radius is bounded by 2 in Section 4.

2 Preliminaries

Let $G=(V,E)$ be a simple graph with vertex set $V=V(G)=\{v_1,v_2,\dots ,v_n\}$ and $e\in E(G)$. Denote by $G-e$ the graph obtained from G by deleting the edge e and by $G-v$ the graph obtained from G by removing the vertex v together with all edges incident to it. For a nonempty subset W of $V(G)$, the subgraph with vertex set W and edge set consisting of those pairs of vertices that are edges in G is called an induced subgraph of G. Denote by $C_n$, $K_{1,n-1}$ and $P_n$ the cycle, the star and the path on n vertices, respectively. Certainly, each subgraph of an oriented graph is also referred as an oriented graph and preserves the orientation of each edge.

Recall that the skew-adjacency matrix $S(G^{\sigma })$ of any oriented graph $G^{\sigma }$ is Hermitian, then the well-known interlacing theorem for Hermitian matrices applies equally well to oriented graphs; see, for example, Theorem 4.3.8 of [17].

Lemma 2.1 Let $G^{\sigma }$ be an arbitrary oriented graph on n vertices and $V^{\prime }\subseteq V(G)$. Suppose that $|V^{\prime }|=k$. Then

$${\lambda }_i\left(G^{\sigma }\right)\ge {\lambda }_i(G^{\sigma }-V^{\prime })\ge {\lambda }_{i+k}\left(G^{\sigma }\right)\mathit{\text{for }}i=1,2,\dots ,n-k.$$

Let $G^{\sigma }$ be an oriented graph, and let $C=v_1{v}_2\cdots v_k{v}_1$ ($k\ge 3$) be a cycle of G, where $v_j$ adjacent to $v_{j+1}$ for $j=1,2,\dots ,k-1$ and $v_1$ adjacent to $v_k$. Let also $S(G^{\sigma })={[s_{ij}]}_{n\times n}$ be the skew-adjacency matrix of $G^{\sigma }$ whose first k rows and columns correspond to the vertices $v_1,v_2,\dots ,v_k$. The sign of the cycle $C^{\sigma }$, denoted by $sgn(C^{\sigma })$, is defined by

$$sgn\left(C^{\sigma }\right)=s_{1,2}{s}_{2,3}\cdots s_{k-1,k}{s}_{k,1}.$$

Let $\overline{C}=v_1{v}_k\cdots v_2{v}_1$ be the cycle by inverting the order of the vertices along the cycle C. Then one can verify that

$$sgn\left({\overline{C}}^{\sigma }\right)=\{\begin{array}{ll}-sgn(C^{\sigma }),& \text{if }k\text{ is odd};\\ sgn(C^{\sigma }),& \text{if }k\text{ is even}.\end{array}$$

Moreover, $sgn(C^{\sigma })$ is either 1 or −1 if the length of C is even; and $sgn(C^{\sigma })$ is either or if the length of C is odd. For an even cycle, we simply refer it as a positive cycle or a negative cycle according to its sign. A positive even cycle is also named as oriented uniformly by Hou et al. [14].

On the skew-spectral radius of an oriented graph, we have obtained the following results. They will be useful in the proofs of the main results of this paper.

Lemma 2.2 ([[16], Theorem 2.1])

Let $G^{\sigma }$ be an arbitrary connected oriented graph. Denote by $\rho (G)$ the (adjacency) spectral radius of G. Then

$${\rho }_s\left(G^{\sigma }\right)\le \rho (G)$$

with equality if and only if G is bipartite and each cycle of G is a positive even cycle.

Lemma 2.3 ([[16], Theorem 3.2])

Let $G^{\sigma }$ be a connected oriented graph. Suppose that each even cycle of G is positive. Then

• (a) ${\rho }_s(G^{\sigma })>{\rho }_s(G^{\sigma }-u)$ for any $u\in G$;

• (b) ${\rho }_s(G^{\sigma })>{\rho }_s(G^{\sigma }-e)$ for any $e\in G$.

Lemma 2.4 ([[14], Theorem 2.4], [[16], Theorem 3.1])

Let $G^{\sigma }$ be an oriented graph, and let $\varphi (G^{\sigma },\lambda )$ be its skew-characteristic polynomial. Then

$$\text{(a)}\varphi (G^{\sigma },\lambda )=\lambda \varphi (G^{\sigma }-u,\lambda )-\sum _{v\in N(u)}\varphi (G^{\sigma }-u-v,\lambda )-2\sum _{u\in C}sgn(C)\varphi (G^{\sigma }-C,\lambda ),$$

where the first summation is over all the vertices in $N(u)$, and the second summation is over all even cycles of G containing the vertex u,

$$\text{(b)}\varphi (G^{\sigma },\lambda )=\varphi (G^{\sigma }-e,\lambda )-\varphi (G^{\sigma }-u-v,\lambda )-2\sum _{(u,v)\in C}sgn(C)\varphi (G^{\sigma }-C,\lambda ),$$

where $e=(u,v)$ and the summation is over all even cycles of G containing the edge e, and $sgn(C)$ denotes the sign of the even cycle C.

Lemma 2.5 ([[13], A part of Theorem 2.5])

Let $G^{\sigma }$ be an oriented graph, and let $\varphi (G^{\sigma },\lambda )$ be its skew-characteristic polynomial. Then

$$\frac{d}{d\lambda }\varphi (G^{\sigma },\lambda )=\sum _{v\in V(G)}\varphi (G^{\sigma }-v,\lambda ),$$

where $\frac{d}{d\lambda }\varphi (G^{\sigma },\lambda )$ denotes the derivative of $\varphi (G^{\sigma },\lambda )$.

Finally, we introduce a class of undirected graphs that will be often mentioned in this manuscript.

Denote by $P_{l_1,l_2,\dots ,l_k}$ a pathlike graph, which is defined as follows: we first draw k (≥2) paths $P_{l_1},P_{l_2},\dots ,P_{l_k}$ of orders $l_1,l_2,\dots ,l_k$ respectively along a line and put two isolated vertices between each pair of those paths, then add edges between the two isolated vertices and the nearest end vertices of such a pair of paths such that the four newly added edges form a cycle $C_4$, where $l_1,l_k\ge 0$ and $l_i\ge 1$ for $i=2,3,\dots ,k-1$. Then $P_{l_1,l_2,\dots ,l_k}$ contains ${\sum }_{i=1}^k{l}_i+2k-2$ vertices. Notice that if $l_i=1$ ($i=2,3,\dots ,k-1$), the two end vertices of the path $P_{l_i}$ are referred as overlap; if $l_1=0$ ($l_k=0$), the left (right) of the graph $P_{l_1,l_2,\dots ,l_k}$ has only two pendent vertices. Obviously, $P_{1,0}=K_{1,2}$, the star of order 3, and $P_{1,1}=C_4$. In general, $P_{l_1,l_2}$, $P_{0,l_1,l_2}$, $P_{0,l_1,l_2,0}$ are all unicyclic graphs containing $C_4$, where $l_1,l_2\ge 1$.

3 The oriented unicyclic graphs whose skew-spectral radius does not exceed 2

In this section, we determine all the oriented unicyclic graphs whose skew-spectral radius does not exceed 2.

First, we introduce more notations. Denote by $T_{l_1,l_2,l_3}$ the starlike tree with exactly one vertex v of degree 3, and $T_{l_1,l_2,l_3}-v=P_{l_1}\cup P_{l_2}\cup P_{l_3}$, where $P_{l_i}$ is the path of order $l_i$ ($i=1,2,3$).

Due to Smith, all undirected graphs whose (adjacency) spectral radius is bounded by 2 are completely determined as follows.

Lemma 3.1 ([2] or [[1], Chapter 2.7.12])

All undirected graphs whose spectral radius does not exceed 2 are $C_m$, $P_{0,n-4,0}$, $T_{2,2,2}$, $T_{1,3,3}$, $T_{1,2,5}$ and their subgraphs, where $m\ge 3$ and $n\ge 5$.

By Lemma 2.4, to study the skew-spectrum properties of an oriented graph, we need only consider the sign of those even cycles. Moreover, Shader and So showed that $S(G^{\sigma })$ has the same spectrum as that of its underlying tree for any oriented tree $G^{\sigma }$; see Theorem 2.5 of [12]. Consequently, combining with Lemma 2.2, the skew-spectral radius of each oriented graph whose underlying graph is as described in Lemma 3.1, regardless of the orientation of the oriented cycle $C_n^{\sigma }$, does not exceed 2.

For convenience, we write:

$\mathcal{U}=\{G|G\text{ is a unicyclic graph}\}$.

$\mathcal{U}(m)=\{G|G\text{ is a unicyclic graph in }\mathcal{U}\text{ containing the cycle }{C}_m\}$.

${\mathcal{U}}^{\ast }(m)=\{G|G\text{ is a unicyclic graph in }\mathcal{U}(m)\text{ which is not the cycle }{C}_m\}$.

${\mathcal{U}}_n=\{G|G\text{ is a unicyclic graph on order }n\}$.

Moreover, let $C_m=v_1{v}_2\cdots v_m{v}_1$ be a cycle on m vertices, and let $P_{l_1},P_{l_2},\dots ,P_{l_m}$ be m paths with lengths $l_1,l_2,\dots ,l_m$ (perhaps some of them are empty), respectively. Denote by $C_m^{l_1,l_2,\dots ,l_m}$ the unicyclic undirected graph obtained from $C_m$ by joining $v_i$ to a pendent vertex of $P_{l_i}$ for $i=1,2,\dots ,m$. Suppose, without loss of generality, that $l_1=max\{l_i:i=1,2,\dots ,m\}$, $l_2\ge l_m$, and write $C_m^{l_1,l_2,\dots ,l_j}$ instead of the standard $C_m^{l_1,l_2,\dots ,l_j,0,\dots ,0}$ if $l_{j+1}=l_{j+2}=\cdots =l_m=0$.

By Lemmas 2.2 and 2.4 or papers [11, 12], for a given unicyclic graph $G\in \mathcal{U}(m)$, we know that the skew-spectral radius of $G^{\sigma }$ is independent of its orientation if m is odd. Therefore, we will briefly write $\overrightarrow{G}$ instead of the normal notation $G^{\sigma }$ if each cycle of G is odd. If m is even, then essentially, there exist two orientations ${\sigma }_1$ (the sign of the even cycle is positive) and ${\sigma }_2$ (the sign of the even cycle is negative) such that ${\rho }_s(G^{{\sigma }_1})=\rho (G)$ and ${\rho }_s(G^{{\sigma }_2})<\rho (G)$. Henceforth, we will briefly write $G^{-}$ (or $G^{+}$) instead of $G^{\sigma }$ if the sign of each even cycle is negative (or positive). In particular, G will also denote the oriented graph if G is a tree since ${\rho }_s(G^{\sigma })=\rho (G)$ in this case.

3.1 The $C_4$-free oriented unicyclic graphs whose skew-spectral radius does not exceed 2

Let $G^{\sigma }$ be an oriented graph with the property

$${\rho }_s\left(G^{\sigma }\right)\le 2.$$

(3.1)

The property (3.1) is hereditary, because, as a direct consequence of Lemma 2.1, for any induced subgraph $H\subset G$, $H^{\sigma }$ also satisfies (3.1). The inheritance (hereditary) of property (3.1) implies that there are minimal connected graphs that do not obey (3.1); such graphs are called forbidden subgraphs. It is easy to verify the following.

Lemma 3.2 Let $G\in \mathcal{U}\setminus \mathcal{U}(4)$ with ${\rho }_s(G^{\sigma })\le 2$. Then ${\overrightarrow{C}}_3^3$, ${\overrightarrow{C}}_3^{1,1}$, ${\overrightarrow{C}}_3(2)$, ${\overrightarrow{C}}_3^1(2)$, ${\overrightarrow{C}}_5^1$, ${\overrightarrow{C}}_7^1$ are forbidden, where ${\overrightarrow{C}}_3(2)$ (or ${\overrightarrow{C}}_3^1(2)$) denotes the oriented graph obtained by adding two pendent vertices to a vertex (or the pendent vertex) of ${\overrightarrow{C}}_3$ (or ${\overrightarrow{C}}_3^1$).

Combining with Lemma 3.2 and the fact that ${\rho }_s(T)>2$ if the oriented tree T contains an arbitrary tree described as Lemma 3.1 as a proper subgraph, we have the following result.

Theorem 3.1 Let $G\in \mathcal{U}\setminus \mathcal{U}(4)$ and $G\ne C_m$. Let also ${\rho }_s(G^{\sigma })\le 2$. Then $G^{\sigma }$ is one of ${\overrightarrow{C}}_3^2$, ${(C_6^{2,0,0,2})}^{-}$, ${(C_6^{1,0,1,0,1})}^{-}$, ${(C_8^{1,0,0,0,1})}^{-}$ and their induced oriented unicyclic subgraphs.

Proof Denote by $gir(G)$ the girth of G. Let $gir(G)=m$ and $C_m$ be the cycle of G with vertex set $\{v_1,v_2,\dots ,v_m\}$ such that $v_i$ adjacent to $v_{i+1}$ for $i=1,2,\dots ,m-1$ and $v_m$ adjacent to $v_1$. (We should point out once again that in $C_m^{l_1,l_2,\dots ,l_j}$ ($j\le m$), we always refer $v_i$ adjacent to one pendent vertex of $P_{l_i}$, a path with length $l_i$, for $i=1,2,\dots ,j$.) We divide our proof into the following four claims.

Claim 1 If $gir(G)=3$, then $G^{\sigma }\in \{{\overrightarrow{C}}_3^1,{\overrightarrow{C}}_3^2\}$.

The result follows from Lemma 3.2 that ${\overrightarrow{C}}_3^3$, ${\overrightarrow{C}}_3^{1,1}$ and ${\overrightarrow{C}}_3(2)$, ${\overrightarrow{C}}_3^1(2)$ are forbidden.

Claim 2 If $gir(G)\ne 3$, then $gir(G)\in \{6,8\}$. Moreover, each induced even cycle of $G^{\sigma }$ is negative.

Let $gir(G)=m$. Notice that G is $C_4$-free, then $m\ge 5$ if $m\ne 3$, and, thus, G contains the induced subgraph $C_m^1$ as $G\ne C_n$. From Lemma 3.2, both ${\overrightarrow{C}}_5^1$ and ${\overrightarrow{C}}_7^1$ are forbidden, thus, $m\ne 5,7$. Moreover, the graph obtained from $C_m^1$ by deleting the vertex $v_5$ is the tree $T_{1,3,m-5}$ for $m\ge 6$. Thus, there is an induced subgraph $T_{1,3,4}$ if $gir(G)\ge 9$, which is a contradiction to Lemma 3.1. Hence, the former follows.

Assume to the contrary that there exists a positive even cycle $C_m^{+}$, then by Lemma 2.3, ${\rho }_s(G^{\sigma })\ge {\rho }_s({(C_m^1)}^{+})>{\rho }_s(C_m^{+})=2$, a contradiction. Thus, the latter follows.

Claim 3 If $gir(G)=6$, then $G^{\sigma }$ is one of ${(C_6^{1,0,1,0,1})}^{-}$, ${(C_6^{2,0,0,2})}^{-}$ or their induced subgraphs.

By Claim 2, we always suppose that each cycle ${\hat{C}}_6$ is negative.

We first claim that G is of $C_6^{l_1,l_2,l_3,l_4,l_5,l_6}$, that is, each pendent tree adjacent to $v_i$ of $C_6$ is a path for $i=1,2,\dots ,6$. Otherwise, assume that the pendent tree adjacent to $v_1$ is not a path, then the resultant graph by deleting vertex $v_3$ of G is a tree and contains the tree $P_{0,l,0}$ as a proper induced subgraph, and, thus, ${\rho }_s(G^{\sigma })>{\rho }_s(P_{0,l,0})=2$ combining with Lemmas 2.3 and 3.2, a contradiction. Moreover, we have $l_1\le 2$. Otherwise, $G-v_4$ contains $T_{2,2,3}$ as an induced subgraph. Notice that both $C_6^{1,1}-v_4$ and $C_6^{2,0,1}-v_5$ are trees and contain $P_{0,2,0}$ as a proper induced subgraph, then G may be $C_6^{1,0,1,0,1}$ and $C_6^{2,0,0,2}$. By calculation, we have ${\rho }_s({(C_6^{1,0,1,0,1})}^{-})=2$ and ${\rho }_s({(C_6^{2,0,0,2})}^{-})=2$. Thus, the result follows.

Claim 4 If $gir(G)=8$, then $G^{\sigma }$ is one of ${(C_8^{1,0,0,0,1})}^{-}$ or its induced subgraphs.

By Claim 2, the cycle $C_8^{\sigma }$ of $G^{\sigma }$ is negative. Notice that $C_8^2-v_5=T_{2,3,3}$, $C_8^{1,1}-v_5=T_{2,2,3}$, $C_8^{1,0,1}-v_5=T_{2,2,3}$, $C_8^{1,0,0,1}-v_5=T_{2,2,3}$, each of them has skew-spectral radius greater than 2. Then $G^{\sigma }$ may be ${(C_8^{1,0,0,0,1})}^{-}$. By calculation, we have ${\rho }_s({(C_8^{1,0,0,0,1})}^{-})=2$. Thus, the result follows.  □

3.2 The oriented unicyclic graphs in $\mathcal{U}(4)$ whose skew-spectral radius does not exceed 2

Now, we consider the oriented unicyclic graphs in $\mathcal{U}(4)$. First, we have the following.

Lemma 3.3 Let $l_1,l_2\ge 1$. Then

• (a) ${\rho }_s(P_{l_1,l_2}^{-})<2$;

• (b) ${\rho }_s(P_{0,l_1,l_2}^{-})={\rho }_s(P_{0,l_1,l_2,0}^{-})=2$.

Proof (a) Let $n=l_1+l_2+2$. We first show by induction on n that

$$\varphi (P_{l_1,l_2}^{-},2)=4.$$

(3.2)

Let $l_1\ge l_2$. Then there is exactly one pathlike graph if $n=4$, namely, $P_{1,1}=C_4$. By calculation, we have

$$\varphi (P_{1,1}^{-},2)=4.$$

Suppose now that $n\ge 5$, and the result is true for the order no more than $n-1$. Applying Lemma 2.4 to the left pendent vertex of $P_{l_1,l_2}^{-}$, we have

$$\varphi (P_{l_1,l_2}^{-},\lambda )=\lambda \varphi (P_{l_1-1,l_2}^{-},\lambda )-\varphi (P_{l_1-2,l_2}^{-},\lambda ).$$

Then $\varphi (P_{l_1,l_2}^{-},2)=4$ by induction hypothesis, and, thus, the result follows.

Let now v be a vertex with degree 2 in $C_4$ of $P_{l_1,l_2}$. Then $P_{l_1,l_2}-v=P_{n-1}$, a path of order $n-1$. Let ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$ and ${\overline{\lambda }}_1\ge {\overline{\lambda }}_2\ge \cdots \ge {\overline{\lambda }}_{n-1}$ be all eigenvalues of $P_{l_1,l_2}^{-}$ and $P_{n-1}$, respectively. By Lemma 2.1 and the fact that ${\overline{\lambda }}_1<2$, we have ${\lambda }_2\le {\overline{\lambda }}_1<2$. On the other hand, we have

$$\varphi (P_{l_1,l_2}^{-},\lambda )=\prod _{i=1}^n(\lambda -{\lambda }_i).$$

Consequently, ${\lambda }_1<2$, and, thus, ${\rho }_s(P_{l_1,l_2}^{-})<2$ by Eq. (3.2). Thus, the result (a) holds.

• (b) We first show that 2 is an eigenvalue of $P_{0,l_1,l_2}^{-}$.

  $$\varphi (P_{0,l_1,l_2}^{-},\lambda )=\lambda \varphi (P_{l_1+1,l_2}^{-},\lambda )-\lambda \varphi (P_{l_1-1,l_2}^{-},\lambda ).$$

By the proof of the result (a), we know that

$$\varphi (P_{l_1+1,l_2}^{-},2)=\varphi (P_{l_1-1,l_2}^{-},2)=4.$$

It tells us that

$$\varphi (P_{0,l_1+1,l_2}^{-},2)=0.$$

Note that ${\lambda }_2(P_{0,l_1+1,l_2}^{-})<2$. We know that ${\rho }_s(P_{0,l_1,l_2}^{-})=2$.

Now, we show that 2 is also an eigenvalue of $P_{0,l_1,l_2,0}^{-}$. Applying Lemma 2.5, we have

$$\frac{d}{d\lambda }\varphi (P_{0,l_1,l_2,0}^{-},\lambda )=\sum _v\varphi (P_{0,l_1,l_2,0}^{-}-v,\lambda ).$$

It is easy to see that 2 is an eigenvalue of each oriented graph $P_{0,l_1,l_2,0}^{-}-v$. Thus, 2 is an eigenvalue of $P_{0,l_1,l_2,0}^{-}$ with multiplicity 2. □

By calculation, we have the following.

Lemma 3.4 Let $G\in \mathcal{U}(4)$ with ${\rho }_s(G^{\sigma })\le 2$. Then $G_i^{-}$ ($i=1,2,\dots ,7$) are forbidden, where $G_1=C_4^{5,1}$, $G_2=C_4^{3,2}$, $G_3=C_4^{4,1,1}$, $G_4=C_4^{3,1,2}$, $G_5=C_4^{2,2,1}$, $G_6=C_4^{3,1,0,1}$ and $G_7=P_{0,l_1,l_2,0}^1$, which denotes the graph obtained by adding a pendent vertex to a vertex of $P_{0,l_1,l_2,0}$.

Combining with Lemma 3.4 and the fact that ${\rho }_s(T)>2$ if the oriented tree T contains an arbitrary tree described as Lemma 3.1 as a proper subgraph, we have the following result.

Theorem 3.2 Let $G\in {\mathcal{U}}^{\ast }(4)$ and ${\rho }_s(G^{\sigma })\le 2$. Then $G^{\sigma }$ is one of ${(C_4^{4,1})}^{-}$, ${(C_4^{3,1,1})}^{-}$, ${(C_4^{2,1,2,1})}^{-}$, ${(C_4^{2,2})}^{-}$ and $P_{0,l_1,l_2,0}^{-}$ or their induced oriented unicyclic subgraphs.

Proof Note that the induced cycle $C_4^{\sigma }$ of $G^{\sigma }$ must be negative. By Lemma 3.3, we can assume that $G\ne P_{0,l_1,l_2,0}$.

Case 1. $G\ne C_4^{l_1,l_2,l_3,l_4}$.

Then G contains an induced tree T such that T has a proper induced subgraph $P_{0,l,0}$. It means that ${\rho }_s(G^{\sigma })>\rho (P_{0,l,0})=2$, a contradiction.

Case 2. $G=C_4^{l_1,l_2,l_3,l_4}$.

Then, by Lemma 3.4, we know that $l_1\le 4$ and $l_2\ge 1$. Thus, it is not difficult to see that the possible oriented graphs are ${(C_4^{4,1})}^{-}$, ${(C_4^{3,1,1})}^{-}$, ${(C_4^{2,1,2,1})}^{-}$, ${(C_4^{2,2})}^{-}$ or their induced oriented unicyclic subgraphs by Lemma 3.4. Moreover, taking some computations, we know the skew-spectral radius of each above oriented graph does not exceed 2.

Combining with Lemma 3.3, the result follows. □

3.3 The oriented unicyclic graphs whose skew-spectral radius does not exceed 2

Putting Lemma 3.1 together with Theorem 3.1 and Theorem 3.2, we have the following.

Theorem 3.3 Let $G\in \mathcal{U}$ and ${\rho }_s(G^{\sigma })\le 2$. Then $G^{\sigma }$ is one of $C_m^{\sigma }$, ${\overrightarrow{C}}_3^2$, ${(C_4^{4,1})}^{-}$, ${(C_4^{3,1,1})}^{-}$, ${(C_4^{2,1,2,1})}^{-}$, ${(C_4^{2,2})}^{-}$, ${(C_6^{2,0,0,2})}^{-}$, ${(C_6^{1,0,1,0,1})}^{-}$, ${(C_8^{1,0,0,0,1})}^{-}$ and $P_{0,l_1,l_2,0}^{-}$ or their induced oriented unicyclic subgraphs, where the orientation of $C_m^{\sigma }$ is arbitrary.

Moreover, by calculation, we have the following two corollaries from Theorem 3.3.

Corollary 3.1 Let $G\in \mathcal{U}$ and ${\rho }_s(G^{\sigma })=2$. Then $G^{\sigma }$ is one of the following oriented graphs.

• (a) $C_m^{+}$, where m is even;

• (b) $P_{0,l_1,l_2}^{-}$, $P_{0,l_1,l_2,0}^{-}$, where $l_1,l_2\ge 1$;

• (c) ${\overrightarrow{C}}_3^2$, ${(C_4^{4,1})}^{-}$, ${(C_4^{3,1,1})}^{-}$, ${(C_4^{2,1,2,1})}^{-}$, ${(C_4^{2,1,2})}^{-}$, ${(C_4^{2,1,1,1})}^{-}$, ${(C_4^{2,1,0,1})}^{-}$, ${(C_4^{2,2})}^{-}$, ${(C_6^{2,0,0,2})}^{-}$, ${(C_6^{1,0,1,0,1})}^{-}$, ${(C_8^{1,0,0,0,1})}^{-}$ and ${(C_8^1)}^{-}$.

Corollary 3.2 Let $G\in \mathcal{U}$ and ${\rho }_s(G^{\sigma })<2$. Then $G^{\sigma }$ is one of the following oriented graphs or their induced oriented unicyclic subgraphs.

• (a) $C_m^{\sigma }$, where m is odd, or m is even, and the sign of $C_m^{\sigma }$ is negative;

• (b) $P_{l_1,l_2}^{-}$, where $l_1,l_2\ge 1$;

• (c) ${\overrightarrow{C}}_3^1$, ${(C_4^{3,1})}^{-}$, ${(C_4^{2,1,1})}^{-}$, ${(C_4^{1,1,1,1})}^{-}$, ${(C_6^{2,0,0,1})}^{-}$, ${(C_6^{1,0,1})}^{-}$.

4 Ordering the oriented unicyclic graphs whose skew-spectral radius is bounded by 2

In this section, we discuss the skew-spectral radii of oriented unicyclic graphs in ${\mathcal{U}}_n$. Let $G\in {\mathcal{U}}_n$ and ${\rho }_s(G^{\sigma })<2$. By Corollary 3.2, we know that $G^{\sigma }$ is $C_n^{\sigma }$ (where n is odd, or n is even, and the sign is negative) or $P_{l,n-l}^{-}$ (where $l\ge 1$) if $n\ge 10$. This makes it possible to order the oriented unicyclic graphs whose skew-spectral radius is bounded by 2.

Lemma 4.1 Let $l_2\ge l_1\ge 2$. Then ${\rho }_s(P_{l_1,l_2}^{-})<{\rho }_s(P_{l_1-1,l_2+1}^{-})$.

Proof By Lemma 2.4, we have

$$\begin{array}{c}\varphi \left(P_{l_1,l_2}^{-}\right)=\lambda \varphi (P_{l_1+l_2+1})-\varphi (P_{l_1-1})\varphi (P_{l_2+1})-\varphi (P_{l_1+1})\varphi (P_{l_2-1})+2\varphi (P_{l_1-1})\varphi (P_{l_2-1});\hfill \\ \varphi \left(P_{l_1-1,l_2+1}^{-}\right)=\lambda \varphi (P_{l_1+l_2+1})-\varphi (P_{l_1-2})\varphi (P_{l_2+2})-\varphi (P_{l_1})\varphi (P_{l_2})+2\varphi (P_{l_1-2})\varphi (P_{l_2}).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\varphi \left(P_{l_1,l_2}^{-}\right)-\varphi \left(P_{l_1-1,l_2+1}^{-}\right)\hfill \\ =[\varphi (P_{l_1-2})\varphi (P_{l_2+2})-\varphi (P_{l_1-1})\varphi (P_{l_2+1})]+[\varphi (P_{l_1})\varphi (P_{l_2})-\varphi (P_{l_1+1})\varphi (P_{l_2-1})]\hfill \\ +2[\varphi (P_{l_1-1})\varphi (P_{l_2-1})-\varphi (P_{l_1-2})\varphi (P_{l_2})].\hfill \end{array}$$

Moreover, we have

$$\begin{array}{c}\varphi (P_{l_1-2})\varphi (P_{l_2+2})-\varphi (P_{l_1-1})\varphi (P_{l_2+1})\hfill \\ =\varphi (P_{l_1-2})[\lambda \varphi (P_{l_2+1})-\varphi (P_{l_2})]-\varphi (P_{l_2+1})[\lambda \varphi (P_{l_1-2})-\varphi (P_{l_1-3})]\hfill \\ =\varphi (P_{l_1-3})\varphi (P_{l_2+1})-\varphi (P_{l_1-2})\varphi (P_{l_2})\hfill \\ =\varphi (P_0)\varphi (P_{l_2-l_1+4})-\varphi (P_1)\varphi (P_{l_2-l_1+3})\hfill \\ =\varphi (P_{l_2-l_1+4})-\lambda \varphi (P_{l_2-l_1+3})\hfill \\ =-\varphi (P_{l_2-l_1+2}),\hfill \\ \varphi (P_{l_1})\varphi (P_{l_2})-\varphi (P_{l_1+1})\varphi (P_{l_2-1})\hfill \\ =\varphi (P_{l_1})[\lambda \varphi (P_{l_2-1})-\varphi (P_{l_2-2})]-\varphi (P_{l_2-1})[\lambda \varphi (P_{l_1})-\varphi (P_{l_1-1})]\hfill \\ =\varphi (P_{l_1-1})\varphi (P_{l_2-1})-\varphi (P_{l_1})\varphi (P_{l_2-2})\hfill \\ =\varphi (P_0)\varphi (P_{l_2-l_1})-\varphi (P_1)\varphi (P_{l_2-l_1-1})\hfill \\ =\varphi (P_{l_2-l_1})-\lambda \varphi (P_{l_2-l_1-1})\hfill \\ =-\varphi (P_{l_2-l_1-2}),\hfill \end{array}$$

where $l_2-l_1\ge 2$. It is easy to know that

$$\varphi (P_{l_1})\varphi (P_{l_2})-\varphi (P_{l_1+1})\varphi (P_{l_2-1})=1\text{if }{l}_2-l_1=0$$

and

$$\varphi (P_{l_1})\varphi (P_{l_2})-\varphi (P_{l_1+1})\varphi (P_{l_2-1})=0\text{if }{l}_2-l_1=1.$$

Similarly, we have

$$\begin{array}{c}\varphi (P_{l_1-1})\varphi (P_{l_2-1})-\varphi (P_{l_1-2})\varphi (P_{l_2})\hfill \\ =\varphi (P_{l_2-1})[\lambda \varphi (P_{l_1-2})-\varphi (P_{l_1-3})]-\varphi (P_{l_1-2})[\lambda \varphi (P_{l_2-1})-\varphi (P_{l_2-2})]\hfill \\ =\varphi (P_{l_1-2})\varphi (P_{l_2-2})-\varphi (P_{l_1-3})\varphi (P_{l_2-1})\hfill \\ =\varphi (P_1)\varphi (P_{l_2-l_1+1})-\varphi (P_0)\varphi (P_{l_2-l_1+2})\hfill \\ =\lambda \varphi (P_{l_2-l_1})-\varphi (P_{l_2-l_1+1})\hfill \\ =\varphi (P_{l_2-l_1}).\hfill \end{array}$$

Hence,

$$\varphi \left(P_{l_1,l_2}^{-}\right)-\varphi \left(P_{l_1-1,l_2+1}^{-}\right)=-\varphi (P_{l_2-l_1+2})-\varphi (P_{l_2-l_1-2})+2\varphi (P_{l_2-l_1}).$$

Let $l_2-l_1=k$. Then for $k\ge 2$, we have

$$\begin{array}{rcl}\varphi \left(P_{l_1,l_2}^{-}\right)-\varphi \left(P_{l_1-1,l_2+1}^{-}\right)& =& -\varphi (P_{k+2})-\varphi (P_{k-2})+2\varphi (P_k)\\ =& -[\varphi (P_2)\varphi (P_k)-\varphi (P_1)\varphi (P_{k-1})]-\varphi (P_{k-2})+2\varphi (P_k)\\ =& (3-{\lambda }^2)\varphi (P_k)+\varphi (P_1)\varphi (P_{k-1})-\varphi (P_{k-2})\\ =& (4-{\lambda }^2)\varphi (P_k).\end{array}$$

Obviously, the above equality also holds for $k=0,1$. It means that $\varphi (P_{l_1,l_2}^{-},{\rho }_s(P_{l_1-1,l_2+1}^{-}))>0$, since ${\rho }_s(P_{l_1-1,l_2+1}^{-})<2$. Thus, ${\rho }_s(P_{l_1,l_2}^{-})<{\rho }_s(P_{l_1-1,l_2+1}^{-})$. □

By Lemma 4.1, we know that

$${\rho }_s\left(P_{\lfloor \frac{n-2}{2}\rfloor ,\lceil \frac{n-2}{2}\rceil }^{-}\right)<\cdots <{\rho }_s\left(P_{2,n-4}^{-}\right)<{\rho }_s\left(P_{1,n-3}^{-}\right).$$

Now, we need only to compare the skew-spectral radii of $P_{l_1,l_2}^{-}$ and $C_n^{\sigma }$. In fact, we have the following.

Lemma 4.2 Let $n\ge 4$. Then we have

• (a) ${\rho }_s(P_{1,n-3}^{-})<{\rho }_s({\overrightarrow{C}}_n)$ if n is odd;

• (b) ${\rho }_s(P_{\frac{n-2}{2},\frac{n-2}{2}}^{-})={\rho }_s(C_n^{-})$ if n is even.

Proof Note that by paper [11]

$${\rho }_s\left(C_n^{\sigma }\right)=\{\begin{array}{ll}2cos\frac{\pi }{2n},& \text{if }n\text{ is odd};\\ 2cos\frac{\pi }{n},& \text{if }n\text{ is even and the sign of the cycle is negative}.\end{array}$$

Moreover, we have ${\rho }_s(P_{0,n-2})=2cos\frac{\pi }{2n-2}$. Thus, ${\rho }_s({\overrightarrow{C}}_n)>{\rho }_s(P_{0,n-2})$ if n is odd.

On the other hand, we have

$$\begin{array}{c}\varphi \left(P_{1,n-3}^{-}\right)=\lambda \varphi (P_{n-1})-\varphi (P_{n-2})-\varphi (P_2)\varphi (P_{n-4})+2\varphi (P_{n-4});\hfill \\ \varphi (P_{0,n-2})=\lambda \varphi (P_{n-1})-\lambda \varphi (P_{n-3}).\hfill \end{array}$$

Thus,

$$\begin{array}{rcl}\varphi \left(P_{1,n-3}^{-}\right)-\varphi (P_{0,n-2})& =& -\varphi (P_2)\varphi (P_{n-4})+2\varphi (P_{n-4})\\ =& (4-{\lambda }^2)\varphi (P_{n-4}).\end{array}$$

It means that ${\rho }_s(P_{1,n-3}^{-})<{\rho }_s(P_{0,n-2})$. Then the result (a) follows.

If n is even, then let $l=\frac{n-2}{2}$. We have

$$\begin{array}{c}\varphi \left(P_{l,l}^{-}\right)=\lambda \varphi (P_{n-1})-2\varphi (P_{l-1})\varphi (P_{l+1})+2\varphi (P_{l-1})\varphi (P_{l-1});\hfill \\ \varphi \left(C_n^{-}\right)=\lambda \varphi (P_{n-1})-2\varphi (P_{n-2})+2\hfill \\ \phantom{\varphi \left(C_n^{-}\right)}=\lambda \varphi (P_{n-1})-2\varphi (P_{l-1})\varphi (P_{l+1})+2\varphi (P_{l-2})\varphi (P_l)+2.\hfill \end{array}$$

Thus,

$$\begin{array}{rcl}\varphi \left(P_{l,l}^{-}\right)-\varphi \left(C_n^{-}\right)& =& -2\varphi (P_{l-2})\varphi (P_l)+2\varphi (P_{l-1})\varphi (P_{l-1})-2\\ =& 2[\varphi (P_{l-2})\varphi (P_{l-2})-\varphi (P_{l-3})\varphi (P_{l-1})]-2\\ =& 2[\varphi (P_1)\varphi (P_1)-\varphi (P_0)\varphi (P_2)]-2\\ =& 0.\end{array}$$

Then the result (b) holds. □

By Lemmas 4.1 and 4.2, we obtain the following interesting result.

Theorem 4.1 Let $G^{\sigma }$ be an oriented unicyclic graph on order n ($n\ge 10$). $G^{\sigma }\ne P_{l_1,l_2}^{-}$, $C_n^{\sigma }$, where $n=l_1+l_2+2$ and $C_n^{\sigma }=C_n^{-}$ if n is even. Then

• (a) ${\rho }_s(P_{\frac{n-3}{2},\frac{n-1}{2}}^{-})<\cdots <{\rho }_s(P_{1,n-3}^{-})<{\rho }_s({\overrightarrow{C}}_n)<2\le {\rho }_s(G^{\sigma })$ if n is odd;

• (b) ${\rho }_s(C_n^{-})={\rho }_s(P_{\frac{n-2}{2},\frac{n-2}{2}}^{-})<\cdots <{\rho }_s(P_{1,n-3}^{-})<2\le {\rho }_s(G^{\sigma })$ if n is even.

Combining with Corollary 3.1, we have ordered all the oriented unicyclic graphs with n vertices whose skew-spectral radius is bounded by 2.