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

R E S E A R C H

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Ordering the oriented unicyclic graphs whose

skew-spectral radius is bounded by



Ping-Feng Chen

_{, Guang-Hui Xu}

_{and Li-Pu Zhang}

*_{Correspondence:}

ghxu@zafu.edu.cn

2_{School of Science, Zhejiang A&F}

University, Hangzhou, 311300, China

Full list of author information is available at the end of the article

Abstract

LetS(Gσ) be the skew-adjacency matrix of an oriented graphGσwithnvertices, and let

λ

λ

λ

ρ

defined as max{|

λ

λ

λ

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 withnvertices whose skew-spectral radius is bounded by 2.

MSC: 05C50; 15A18

Keywords: oriented unicyclic graph; skew-adjacency matrix; skew-spectral radius

1 Introduction

LetGbe a simple graph withnvertices. Theadjacency matrix A=A(G) is the symmetric matrix [aij]n×n, whereaij=aji=  ifvivjis an edge ofG, otherwise,aij=aji= . We call

det(λI–A) thecharacteristic polynomialofG, denoted byφ(G;λ) (or abbreviated toφ(G)). SinceAis symmetric, its eigenvaluesλ(G),λ(G), . . . ,λn(G) are real, and we assume that λ(G)≥λ(G)≥ · · · ≥λn(G). We callρ(G) =λ(G) theadjacency spectral radiusofG.

The class of all graphs Gwhose largest (adjacency) spectral radius is bounded by  has been completely determined by Smith; see, for example, [, ]. Later, Hoffman [], Cvetkovićet al.[] gave a nearly complete description of all graphsGwith  <ρ(G) <

 +√ (≈.). Their description was completed by Brouwer and Neumaier []. Then Woo and Neumaier [] investigated the structure of graphs Gwith  +√ <

λmax(G) < _ √

 (≈.), Wanget al.[] investigated the structure of graphs whose largest eigenvalue is close to_√.

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 , Guo [] gave the first six unicyclic graphs of ordernwith larger spectral radius. Belardoet al.[] ordered graphs with spectral radius in the interval (, +√). In the paper [], the first five unicyclic graphs on ordernin 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 theoriented graph. LetGσ _{be an oriented graph with vertex set}

toGσ _{is defined as}

sij= ⎧ ⎪ ⎨ ⎪ ⎩

i, if there exists an edge with tailviand headvj;

–i, if there exists an edge with headviand tailvj;

, otherwise,

wherei=√– (note that the definition is slightly different from the one of the normal skew-adjacency matrix given by Adigaet al.[]). SinceS(Gσ_{) is an Hermitian matrix,}

the eigenvaluesλ(Gσ),λ(Gσ), . . . ,λn(Gσ) ofS(Gσ) are all real numbers and, thus, can be

arranged non-increase as

λ

Gσ≥λ

Gσ≥ · · · ≥λn

Gσ.

Theskew-spectral radiusand theskew-characteristic polynomialofGσ _{are defined}

respec-tively as

ρs

Gσ=max λ

Gσ,λ

Gσ, . . . ,λn

Gσ

and

φGσ_;λ=detλI

n–S

Gσ_.

Recently, much attention has been devoted to the skew-adjacency matrix of an oriented graph. In , Shader and So [] investigated the spectra of the skew-adjacency matrix of an oriented graph. In , Adigaet al.[] discussed the properties of the skew-energy of an oriented graph. In papers [, ], all the coefficients of the skew-characteristic poly-nomial ofGσ_{in terms ofGwere interpreted. Caverset al.[] discussed the graphs whose}

skew-adjacency matrices are all cospectral, and the relations between the matchings poly-nomial of a graph and the characteristic polypoly-nomials of its adjacency and skew-adjacency matrices. In [], the author established a relation between ρs(Gσ) andρ(G). Also, the

author gave some results on the skew-spectral radii ofGσ _{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 , we introduce some notations and preliminary results. In Section , all the oriented unicyclic graphs whose skew-spectral radius does not exceed  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 withnvertices whose skew-spectral radius is bounded by  in Section .

2 Preliminaries

LetG= (V,E) be a simple graph with vertex setV =V(G) ={v,v, . . . ,vn}ande∈E(G).

those pairs of vertices that are edges inGis called aninduced subgraphofG. Denote by

Cn,K,n–andPn the cycle, the star and the path onnvertices, 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 matrixS(Gσ_{) of any oriented graphG}σ _{is Hermitian,}

then the well-known interlacing theorem for Hermitian matrices applies equally well to oriented graphs; see, for example, Theorem .. of [].

Lemma . Let Gσ _{be an arbitrary oriented graph on n vertices and V⊆V(G).Suppose}

that|V|=k.Then λi

Gσ≥λi

Gσ–V≥λi+k

Gσ for i= , , . . . ,n–k.

LetGσ _{be an oriented graph, and letC=v}

v· · ·vkv (k≥) be a cycle ofG, wherevj

adjacent tovj+ forj= , , . . . ,k–  andv adjacent tovk. Let alsoS(Gσ) = [sij]n×nbe the

skew-adjacency matrix ofGσ _{whose firstkrows and columns correspond to the vertices}

v,v, . . . ,vk. The sign of the cycleCσ, denoted bysgn(Cσ), is defined by

sgnCσ=s,s,· · ·sk–,ksk,.

LetC¯ =vvk· · ·vv be the cycle by inverting the order of the vertices along the cycleC. Then one can verify that

sgnC¯σ=

–sgn(Cσ_{), ifkis odd;}

sgn(Cσ_{), ifkis even.}

Moreover,sgn(Cσ_{) is either  or – if the length ofCis even; andsgn(C}σ_{) is eitherior –iif}

the length ofCis 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 Houet al.[].

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 .([, Theorem .]) Let Gσ _{be an arbitrary connected oriented graph.Denote}

byρ(G)the(adjacency)spectral radius of G.Then

ρsGσ≤ρ(G)

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

Lemma .([, Theorem .]) Let Gσ _{be a connected oriented graph.Suppose that each}

even cycle of G is positive.Then

(a) ρs(Gσ) >ρs(Gσ –u)for anyu∈G; (b) ρs(Gσ) >ρs(Gσ –e)for anye∈G.

Lemma .([, Theorem .], [, Theorem .]) Let Gσ _{be an oriented graph,and let}

φ(Gσ_{,λ)be its skew-characteristic polynomial.Then}

(a) φGσ,λ=λφGσ –u,λ–

v∈N(u)

φGσ –u–v,λ– 

u∈C

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,

(b) φGσ,λ=φGσ–e,λ–φGσ–u–v,λ–  (u,v)∈C

sgn(C)φGσ–C,λ,

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 .([, A part of Theorem .]) Let Gσ _{be an oriented graph,and letφ(G}σ_,λ)

be its skew-characteristic polynomial.Then

d dλφ

Gσ_,λ=

v∈V(G)

φGσ_–v,λ,

where_dd_λφ(Gσ_{,λ)denotes the derivative ofφ(G}σ_,λ).

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

Denote byPl,l,...,lk a pathlike graph, which is defined as follows: we first drawk(≥) paths Pl,Pl, . . . ,Plk of ordersl,l, . . . ,lk 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 cycleC, where l,lk≥ andli≥ fori= , , . . . ,k– . Then Pl,l,...,lk contains

k

i=li+ k–  vertices. Notice that ifli=  (i= , , . . . ,k– ), the two end vertices of

the pathPliare referred as overlap; ifl=  (lk= ), the left (right) of the graphPl,l,...,lk has only two pendent vertices. Obviously,P,=K,, the star of order , andP,=C. In general,Pl,l,P,l,l,P,l,l,are all unicyclic graphs containingC, wherel,l≥.

3 The oriented unicyclic graphs whose skew-spectral radius does not exceed2 In this section, we determine all the oriented unicyclic graphs whose skew-spectral radius does not exceed .

First, we introduce more notations. Denote byTl,l,l the starlike tree with exactly one

vertexvof degree , andTl,l,l–v=Pl∪Pl∪Pl, wherePliis the path of orderli(i= , , ).

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

Lemma .([] or [, Chapter ..]) All undirected graphs whose spectral radius does not exceedare Cm,P,n–,,T,,,T,,,T,,and their subgraphs,where m≥and n≥.

By Lemma ., 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 thatS(Gσ₎

has the same spectrum as that of its underlying tree for any oriented treeGσ_{; see}

The-orem . of []. Consequently, combining with Lemma ., the skew-spectral radius of each oriented graph whose underlying graph is as described in Lemma ., regardless of the orientation of the oriented cycleCσ

For convenience, we write: U={G|Gis a unicyclic graph}.

U(m) ={G|Gis a unicyclic graph inUcontaining the cycleCm}.

U∗_{(m) ={G|Gis a unicyclic graph inU(m)which is not the cycleC}

m}.

Un={G|Gis a unicyclic graph on ordern}.

Moreover, letCm=vv· · ·vmv be a cycle onmvertices, and letPl,Pl, . . . ,Plm bem paths with lengthsl,l, . . . ,lm(perhaps some of them are empty), respectively. Denote by Cl,l,...,lm

m the unicyclic undirected graph obtained fromCmby joiningvito a pendent vertex

ofPlifori= , , . . . ,m. Suppose, without loss of generality, thatl=max{li:i= , , . . . ,m},

l≥lm, and writeC l,l,...,lj

m instead of the standardC

l,l,...,lj,,...,

m iflj+=lj+=· · ·=lm= .

By Lemmas . and . or papers [, ], for a given unicyclic graphG∈U(m), we know that the skew-spectral radius ofGσ _{is independent of its orientation ifmis odd. Therefore,}

we will briefly writeGinstead of the normal notationGσ _{if each cycle ofGis odd. Ifmis}

even, then essentially, there exist two orientationsσ(the sign of the even cycle is positive) andσ(the sign of the even cycle is negative) such thatρs(Gσ) =ρ(G) andρs(Gσ) <ρ(G). Henceforth, we will briefly writeG–_(orG+_{) instead ofG}σ _{if the sign of each even cycle}

is negative (or positive). In particular,Gwill also denote the oriented graph ifGis a tree sinceρs(Gσ) =ρ(G) in this case.

3.1 TheC4-free oriented unicyclic graphs whose skew-spectral radius does not exceed 2

LetGσ _{be an oriented graph with the property}

ρs

Gσ_{≤. (.)}

The property (.) is hereditary, because, as a direct consequence of Lemma ., for any induced subgraphH⊂G,Hσ _{also satisfies (.). The inheritance (hereditary) of property}

(.) implies that there are minimal connected graphs that do not obey (.); such graphs are calledforbidden subgraphs. It is easy to verify the following.

Lemma . Let G∈U\U()withρs(Gσ)≤.ThenC,C ,

 ,C(),C(),C,Care

for-bidden,whereC() (orC())denotes the oriented graph obtained by adding two pendent

vertices to a vertex(or the pendent vertex)ofC(orC).

Combining with Lemma . and the fact thatρs(T) >  if the oriented treeTcontains an

arbitrary tree described as Lemma . as a proper subgraph, we have the following result.

Theorem . Let G∈U\U()and G=Cm.Let alsoρs(Gσ)≤.Then Gσ is one ofC, (C_,,,)–_{, (C},,,,

 )–, (C ,,,,

 )–and their induced oriented unicyclic subgraphs.

Proof Denote bygir(G) the girth ofG. Letgir(G) =mandCmbe the cycle ofGwith vertex

set{v,v, . . . ,vm}such thatviadjacent tovi+fori= , , . . . ,m–  andvmadjacent tov. (We should point out once again that inCml,l,...,lj(j≤m), we always referviadjacent to one

pendent vertex ofPli, a path with lengthli, fori= , , . . . ,j.) We divide our proof into the following four claims.

The result follows from Lemma . thatC_,C_,andC(),C() are forbidden.

Claim  Ifgir(G)= ,thengir(G)∈ {, }.Moreover,each induced even cycle of Gσ _is

neg-ative.

Letgir(G) =m. Notice thatGisC-free, thenm≥ ifm= , and, thus,Gcontains the induced subgraphC

masG=Cn. From Lemma ., bothCandCare forbidden, thus,

m= , . Moreover, the graph obtained fromC

mby deleting the vertexvis the treeT,,m– form≥. Thus, there is an induced subgraphT,,ifgir(G)≥, which is a contradiction to Lemma .. Hence, the former follows.

Assume to the contrary that there exists a positive even cycleC+

m, then by Lemma ., ρs(Gσ)≥ρs((Cm)+) >ρs(Cm+) = , a contradiction. Thus, the latter follows.

Claim  Ifgir(G) = ,then Gσ _{is one of(C},,,,  )–, (C

,,,

 )–or their induced subgraphs.

By Claim , we always suppose that each cycleCis negative. We first claim thatGis ofCl,l,l,l,l,l

 , that is, each pendent tree adjacent toviofCis a path fori= , , . . . , . Otherwise, assume that the pendent tree adjacent tovis not a path, then the resultant graph by deleting vertexvofGis a tree and contains the treeP,l,as a proper induced subgraph, and, thus,ρs(Gσ) >ρs(P,l,) =  combining with Lemmas . and ., a contradiction. Moreover, we havel≤. Otherwise,G–vcontainsT,,as an induced subgraph. Notice that bothC_,–vandC,,–vare trees and containP,,as a proper induced subgraph, thenGmay beC_,,,,andC,,,_ . By calculation, we have

ρs((C_,,,,)–_{) =  andρs((C},,,

 )–) = . Thus, the result follows.

Claim  Ifgir(G) = ,then Gσ _{is one of(C},,,,

 )–or its induced subgraphs.

By Claim , the cycleCσ

 ofGσ is negative. Notice thatC–v=T,,,C,–v=T,,,

C_,,–v=T,,,C,,,–v=T,,, each of them has skew-spectral radius greater than . ThenGσ _{may be (C},,,,

 )–. By calculation, we haveρs((C,,,,)–) = . Thus, the result

follows.

3.2 The oriented unicyclic graphs inU(4) whose skew-spectral radius does not exceed 2

Now, we consider the oriented unicyclic graphs inU(). First, we have the following.

Lemma . Let l,l≥.Then

(a) ρs(P–

l,l) < ;

(b) ρs(P–

,l,l) =ρs(P

–

,l,l,) = .

Proof (a) Letn=l+l+ . We first show by induction onnthat

φP–_l,l, = . (.)

Letl≥l. Then there is exactly one pathlike graph ifn= , namely,P,=C. By calcula-tion, we have

Suppose now thatn≥, and the result is true for the order no more thann– . Applying Lemma . to the left pendent vertex ofP–

l,l, we have

φP–_l,l,λ=λφP–_{l–,l},λ–φP–_{l–,l},λ.

Thenφ(P–

l,l, ) =  by induction hypothesis, and, thus, the result follows.

Let nowvbe a vertex with degree  inCofPl,l. ThenPl,l–v=Pn–, a path of order

n– . Letλ≥λ≥ · · · ≥λnandλ¯≥ ¯λ≥ · · · ≥ ¯λn–be all eigenvalues ofP–l,l andPn–,

respectively. By Lemma . and the fact thatλ¯< , we have λ≤ ¯λ< . On the other hand, we have

φP–_l,l,λ=

n

i= (λ–λi).

Consequently,λ< , and, thus,ρs(P–l,l) <  by Eq. (.). Thus, the result (a) holds.

(b) We first show that  is an eigenvalue ofP– ,l,l.

φP–_{,l,l},λ=λφP–_{l+,l},λ–λφP–_{l–,l},λ.

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

φP–_{l+,l}, =φP_l–_{–,l}, = .

It tells us that

φP– ,l+,l, 

= .

Note thatλ(P–,l+,l) < . We know thatρs(P

–

,l,l) = .

Now, we show that  is also an eigenvalue ofP–

,l,l,. Applying Lemma ., we have

d dλφ

P_,–_{l,l,},λ=

v

φP_,–_{l,l,}–v,λ.

It is easy to see that  is an eigenvalue of each oriented graphP–

,l,l,–v. Thus,  is an

eigenvalue ofP–_{,l,l,}with multiplicity . By calculation, we have the following.

Lemma . Let G∈U()withρs(Gσ)≤.Then G–i (i= , , . . . , )are forbidden,where G=C, ,G=C,,G=C,, ,G=C,, ,G=C,, ,G=C,,,and G=P,l,l,,which

denotes the graph obtained by adding a pendent vertex to a vertex of P,l,l,.

Combining with Lemma . and the fact thatρs(T) >  if the oriented treeTcontains an

arbitrary tree described as Lemma . as a proper subgraph, we have the following result.

Theorem . Let G∈U∗()andρs(Gσ)≤.Then Gσ is one of(C,)–, (C,,)–, (C,,,)–, (C_,)–_{and P}–

Proof Note that the induced cycleCσ

 of Gσ must be negative. By Lemma ., we can assume thatG=P,l,l,.

Case .G=Cl,l,l,l

 .

ThenGcontains an induced treeTsuch thatThas a proper induced subgraphP,l,. It means thatρs(Gσ) >ρ(P,l,) = , a contradiction.

Case .G=Cl,l,l,l

 .

Then, by Lemma ., we know thatl≤ andl≥. Thus, it is not difficult to see that the possible oriented graphs are (C,_ )–_{, (C},,

 )–, (C ,,,  )–, (C

,

 )–or their induced oriented unicyclic subgraphs by Lemma .. Moreover, taking some computations, we know the skew-spectral radius of each above oriented graph does not exceed .

Combining with Lemma ., the result follows. 3.3 The oriented unicyclic graphs whose skew-spectral radius does not exceed 2 Putting Lemma . together with Theorem . and Theorem ., we have the following.

Theorem . Let G∈U and ρs(Gσ)≤.Then Gσ is one of Cmσ, C, (C ,  )–, (C

,,  )–, (C_,,,)–_{, (C},

 )–, (C,,,)–, (C ,,,,  )–, (C

,,,,

 )– and P–,l,l,or their induced oriented

unicyclic subgraphs,where the orientation of Cσ

mis arbitrary.

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

Corollary . Let G∈Uandρs(Gσ) = .Then Gσ is one of the following oriented graphs. (a) C+

m,wheremis even; (b) P–

,l,l,P

–

,l,l,,wherel,l≥;

(c) C,(C,)–,(C,,)–,(C,,,)–,(C,,)–,(C,,,)–,(C,,,)–,(C, )–,(C,,,)–, (C_,,,,)–_,(C,,,,

 )–and(C)–.

Corollary . Let G∈U andρs(Gσ) < .Then Gσ is one of the following oriented graphs or their induced oriented unicyclic subgraphs.

(a) Cσ

m,wheremis odd,ormis even,and the sign ofCmσ is negative; (b) P–

l,l,wherel,l≥;

(c) C

,(C,)–,(C,,)–,(C,,,)–,(C,,,)–,(C ,,  )–.

4 Ordering the oriented unicyclic graphs whose skew-spectral radius is bounded by2

In this section, we discuss the skew-spectral radii of oriented unicyclic graphs inUn. Let G∈Unandρs(Gσ) < . By Corollary ., we know thatGσ isCnσ (wherenis odd, ornis

even, and the sign is negative) orP–

l,n–l(wherel≥) ifn≥. This makes it possible to

order the oriented unicyclic graphs whose skew-spectral radius is bounded by .

Lemma . Let l≥l≥.Thenρs(P–l,l) <ρs(P

–

l–,l+).

Proof By Lemma ., we have

φP–_l,l=λφ(Pl+l+) –φ(Pl–)φ(Pl+) –φ(Pl+)φ(Pl–) + φ(Pl–)φ(Pl–);

Thus,

φP–_l,l–φP–_{l–,l+}

=φ(Pl–)φ(Pl+) –φ(Pl–)φ(Pl+)

+φ(Pl)φ(Pl) –φ(Pl+)φ(Pl–)

+ φ(Pl–)φ(Pl–) –φ(Pl–)φ(Pl)

.

Moreover, we have

φ(Pl–)φ(Pl+) –φ(Pl–)φ(Pl+)

=φ(Pl–)

λφ(Pl+) –φ(Pl)

–φ(Pl+)

λφ(Pl–) –φ(Pl–)

=φ(Pl–)φ(Pl+) –φ(Pl–)φ(Pl)

=φ(P)φ(Pl–l+) –φ(P)φ(Pl–l+)

=φ(Pl–l+) –λφ(Pl–l+)

= –φ(Pl–l+),

φ(Pl)φ(Pl) –φ(Pl+)φ(Pl–)

=φ(Pl)

λφ(Pl–) –φ(Pl–)

–φ(Pl–)

λφ(Pl) –φ(Pl–)

=φ(Pl–)φ(Pl–) –φ(Pl)φ(Pl–)

=φ(P)φ(Pl–l) –φ(P)φ(Pl–l–)

=φ(Pl–l) –λφ(Pl–l–)

= –φ(Pl–l–),

wherel–l≥. It is easy to know that

φ(Pl)φ(Pl) –φ(Pl+)φ(Pl–) =  ifl–l= 

and

φ(Pl)φ(Pl) –φ(Pl+)φ(Pl–) =  ifl–l= .

Similarly, we have

φ(Pl–)φ(Pl–) –φ(Pl–)φ(Pl)

=φ(Pl–)

λφ(Pl–) –φ(Pl–)

–φ(Pl–)

λφ(Pl–) –φ(Pl–)

=φ(Pl–)φ(Pl–) –φ(Pl–)φ(Pl–)

=φ(P)φ(Pl–l+) –φ(P)φ(Pl–l+)

=λφ(Pl–l) –φ(Pl–l+)

=φ(Pl–l).

Hence,

Letl–l=k. Then fork≥, we have

φP–_l,l–φP–_{l–,l+}= –φ(Pk+) –φ(Pk–) + φ(Pk)

= –φ(P)φ(Pk) –φ(P)φ(Pk–)

–φ(Pk–) + φ(Pk)

= –λφ(Pk) +φ(P)φ(Pk–) –φ(Pk–)

= –λφ(Pk).

Obviously, the above equality also holds fork= , . It means thatφ(P_l–_,l,ρs(P–l–,l+)) >

, sinceρs(Pl––,l+) < . Thus,ρs(P

–

l,l) <ρs(P

–

l–,l+).

By Lemma ., we know that

ρs

P–n–  ,n–

<· · ·<ρs

P_,–_n–<ρs

P–_,n–.

Now, we need only to compare the skew-spectral radii ofP–

l,l andC

σ

n. In fact, we have

the following.

Lemma . Let n≥.Then we have (a) ρs(P,–n–) <ρs(Cn)ifnis odd; (b) ρs(P–n–

 ,n–

) =ρs(Cn–)ifnis even.

Proof Note that by paper []

ρs

Cσ

n

=

cos_π_n, ifnis odd;

cosπ_n, ifnis even and the sign of the cycle is negative.

Moreover, we haveρs(P,n–) = cos_nπ_–. Thus,ρs(Cn) >ρs(P,n–) ifnis odd. On the other hand, we have

φP–_,n–=λφ(Pn–) –φ(Pn–) –φ(P)φ(Pn–) + φ(Pn–);

φ(P,n–) =λφ(Pn–) –λφ(Pn–).

Thus,

φP–_,n––φ(P,n–) = –φ(P)φ(Pn–) + φ(Pn–)

= –λφ(Pn–).

It means thatρs(P–

,n–) <ρs(P,n–). Then the result (a) follows. Ifnis even, then letl=n–

 . We have

φP–_l,l=λφ(Pn–) – φ(Pl–)φ(Pl+) + φ(Pl–)φ(Pl–);

φC_n–=λφ(Pn–) – φ(Pn–) + 

Thus,

φP–_l,l–φC_n–= –φ(Pl–)φ(Pl) + φ(Pl–)φ(Pl–) – 

= φ(Pl–)φ(Pl–) –φ(Pl–)φ(Pl–)

– 

= φ(P)φ(P) –φ(P)φ(P)

– 

= .

Then the result (b) holds.

By Lemmas . and ., we obtain the following interesting result.

Theorem . Let Gσ _{be an oriented unicyclic graph on order n(n≥).G}σ _=P–

l,l,C

σ

n, where n=l+l+ and Cnσ =Cn–if n is even.Then

(a) ρs(P–n–  ,n–

) <· · ·<ρs(P–,n–) <ρs(Cn) < ≤ρs(Gσ)ifnis odd; (b) ρs(Cn–) =ρs(P–n–

 ,n–

) <· · ·<ρs(P–,n–) < ≤ρs(Gσ)ifnis even.

Combining with Corollary ., we have ordered all the oriented unicyclic graphs withn

vertices whose skew-spectral radius is bounded by .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

PC carried out the studies of the skew-spectral radii and drafted the manuscript. GX conceived of the study and finished the final manuscript. LZ participated in the design of the study and some calculation. All authors read and approved the final manuscript.

Author details

1_{School of Information Engineering, Zhejiang A&F University, Hangzhou, 311300, China.}2_{School of Science, Zhejiang}

A&F University, Hangzhou, 311300, China.

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).

Received: 24 March 2013 Accepted: 23 September 2013 Published:07 Nov 2013 References

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Cite this article as:Chen et al.:Ordering the oriented unicyclic graphs whose skew-spectral radius is bounded by2.