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

- Ping-Feng Chen
^{1}, - Guang-Hui Xu
^{2}Email author and - Li-Pu Zhang
^{2}

**2013**:495

https://doi.org/10.1186/1029-242X-2013-495

© Chen et al.; licensee Springer. 2013

**Received:**24 March 2013**Accepted:**23 September 2013**Published:**7 November 2013

## 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.

## Keywords

- oriented unicyclic graph
- skew-adjacency matrix
- skew-spectral radius

## 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 $\sqrt{2+\sqrt{5}}<{\lambda}_{\mathrm{max}}(G)<\frac{3}{2}\sqrt{2}$ (≈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.

*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

*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

*skew-spectral radius*and the

*skew-characteristic polynomial*of ${G}^{\sigma}$ are defined respectively as

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*

*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

*C*. Then one can verify that

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*

*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*

*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*,

*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*

*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

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 ${\stackrel{\u02c6}{C}}_{6}$ 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

Then $\varphi ({P}_{{l}_{1},{l}_{2}}^{-},2)=4$ by induction hypothesis, and, thus, the result follows.

*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

- (b)We first show that 2 is an eigenvalue of ${P}_{0,{l}_{1},{l}_{2}}^{-}$.$\varphi ({P}_{0,{l}_{1},{l}_{2}}^{-},\lambda )=\lambda \varphi ({P}_{{l}_{1}+1,{l}_{2}}^{-},\lambda )-\lambda \varphi ({P}_{{l}_{1}-1,{l}_{2}}^{-},\lambda ).$

Note that ${\lambda}_{2}({P}_{0,{l}_{1}+1,{l}_{2}}^{-})<2$. We know that ${\rho}_{s}({P}_{0,{l}_{1},{l}_{2}}^{-})=2$.

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

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, ${\rho}_{s}({P}_{{l}_{1},{l}_{2}}^{-})<{\rho}_{s}({P}_{{l}_{1}-1,{l}_{2}+1}^{-})$. □

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]

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.

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

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

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.

## Declarations

### Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions, which led to a great improvement of the original manuscript. This work was supported by the National Natural Science Foundation of China (No. 11171373), the Zhejiang Provincial Natural Science Foundation of China (LY12A01016).

## Authors’ Affiliations

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