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The oriented bicyclic graphs whose skewspectral radii do not exceed 2
Journal of Inequalities and Applications volume 2015, Article number: 326 (2015)
Abstract
Let \(S(G^{\sigma})\) be the skewadjacency matrix of the oriented graph \(G^{\sigma}\) on order n and \(\lambda _{1},\lambda _{2},\ldots, \lambda _{n}\) be all eigenvalues of \(S(G^{\sigma})\). The skewspectral radius \(\rho_{s}(G^{\sigma})\) of \(G^{\sigma}\) is defined as \(\max\{\lambda _{1},\lambda _{2},\ldots,\lambda _{n}\}\). In this paper, we determine all the oriented bicyclic graphs whose skewspectral radii do not exceed 2.
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 \(\operatorname{det}(\lambda IA)\) the characteristic polynomial of G, denoted by \(\phi(G; \lambda)\). Since A is symmetric, its eigenvalues \(\lambda_{1}(G), \lambda_{2}(G), \ldots, \lambda_{n}(G)\) are real, and we assume that \(\lambda_{1}(G)\geq\lambda_{2}(G)\geq \cdots\geq\lambda_{n}(G)\). We call \(\rho(G)=\lambda_{1}(G)\) the adjacency spectral radius of G.
The class of all graphs G whose largest (adjacency) eigenvalue \(\lambda _{\max}(G)\) 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 \(2<\lambda _{\max}(G)< \sqrt{2 + \sqrt{5}}\ (\approx 2.0582)\). Their description was completed by Brouwer and Neumaier [5]. And Belardo et al. [6] ordered graphs with spectral radius in the interval \((2, \sqrt{2+\sqrt{5}})\). Then Woo and Neumaier [7] investigated the structure of graphs G with \(\sqrt{2 + \sqrt{5}}<\lambda _{\max}(G)< \frac{3}{2}\sqrt{2}\ (\approx2.1312)\), Wang et al. [8] investigated the structure of graphs whose largest eigenvalue is close to \(\frac{3}{2}\sqrt{2}\). In the paper [9], the first three bicyclic graphs on order n in terms of their larger spectral radii were determined.
The graph obtained from a simple undirected graph by assigning an orientation to each of its edges is referred to as the oriented graph. Let \(G^{\sigma}\) be an oriented graph with a vertex set \(\{v_{1},v_{2},\ldots ,v_{n}\}\) and an edge set \(E(G^{\sigma})\). The skewadjacency matrix \(S=S(G^{\sigma})=[s_{ij}]_{n\times n}\) related to \(G^{\sigma}\) is defined as:
where \(\mathbb {i}=\sqrt{1}\) (note that the definition is slightly different from the one of the normal skewadjacency matrix given by Adiga et al. [10]). Since \(S(G^{\sigma})\) is a Hermitian matrix, the eigenvalues \(\lambda _{1}(G^{\sigma}), \lambda _{2}(G^{\sigma}), \ldots, \lambda _{n}(G^{\sigma})\) of \(S(G^{\sigma})\) are all real numbers and thus can be arranged nonincrease as
The skewspectral radius and the skewcharacteristic polynomial of \(G^{\sigma}\) are defined respectively as
and
We denote by \(\mathfrak{D}(G)\) the set of all the oriented graphs obtained from G by giving an arbitrary orientation to each edge. Recently, much attention has been devoted to the skewadjacency matrix of an oriented graph. In 2009, Shader and So [11] investigated the spectra of the skewadjacency matrix of an oriented graph. And in 2010, Adiga et al. [10] discussed the properties of the skewenergy of an oriented graph. In the papers [12, 13], all the coefficients of the skewcharacteristic polynomial of \(G^{\sigma}\) in terms of G were interpreted. Cavers et al. [14] discussed the graphs whose skewadjacency matrices are all cospectral and the relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skewadjacency matrices. In [15], the author established a relation between \(\rho_{s}(G^{\sigma})\) and \(\rho(G)\). Also, the author gave some results on the skewspectral radii of \(G^{\sigma}\) and its oriented subgraphs. In the paper [16], the oriented graphs whose skewspectral radii do not exceed 2 were investigated. In 2013, Chen et al. [17] ordered all the oriented unicyclic graphs with n vertices whose skewspectral radii are bounded by 2.
A connected graph in which the number of edges equals the number of vertices plus one is called a bicyclic graph. In this paper, we will determine all the oriented bicyclic graphs whose skewspectral radii do not exceed 2. The rest of the paper is organized as follows. In Section 2, we introduce some notations and preliminary results. In Section 3, we give some earlier results on the oriented graphs whose skewspectral radii do not exceed 2. In Section 4, we determine all the oriented bicyclic graphs whose skewspectral radii do not exceed 2.
Preliminaries
Let \(G=(V,E)\) be a simple graph with a vertex set \(V=V(G)=\{v_{1},v_{2},\ldots,v_{n}\}\) and \(e\in E(G)\). A graph H is called a subgraph of a graph G if \(V (H) \in V (G)\) and \(E(H) \in E(G)\). Further, H is called an induced subgraph of G if two vertices of \(V(H)\) are adjacent in H if and only if they are adjacent in G. Denote by \(G  U\), where \(U \in V (G)\), the graph obtained from G by removing the vertices of U together with all edges incident to them. Denote by \(G e\) the subgraph obtained from G by deleting the edge e. We refer to [1, 18, 19] for more terminology and notation not defined here. Certainly, each subgraph of an oriented graph is referred to as an oriented graph and preserves the orientation of each edge.
Recall that the skewadjacency matrix \(S(G^{\sigma})\) of any oriented graph \(G^{\sigma}\) is Hermitian, then the wellknown interlacing theorem for Hermitian matrices applies equally well to oriented graphs; see, for example, Theorem 4.3.8 of [19].
Lemma 2.1
Let \(G^{\sigma}\) be an arbitrary oriented graph on n vertices and \(V' \subseteq V(G)\). Suppose that \(V'=k\). Then
Let \(G^{\sigma}\in\mathfrak{D}(G)\) and \(W=u_{1}u_{2}\cdots u_{k}u_{k+1}\) be a walk of G. The sign of \(W^{\sigma}\) in \(G^{\sigma}\), denoted by \(\operatorname{sgn}(W^{\sigma})\), is defined by
Let \(\bar{W}=u_{k+1}u_{k} \cdots u_{2}u_{1}\) be the walk by inverting the order of the vertices along the walk W. Then one can find that
Obviously, for an even closed walk (that is to say \(u_{k+1}=u_{1}\)), we can simply refer to it as a positive (or negative) even closed walk according to its sign, regardless of the order of its vertices. Similarly, we can define a positive (or negative) even cycle.
We now list some results related to this paper.
Lemma 2.2
([15], 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 \(G^{\sigma}\) is positive.
Lemma 2.3
([13], Theorem 2.4, [15], Theorem 3.1)
Let \(G^{\sigma}\) be an oriented graph and \(\phi(G^{\sigma}, \lambda )\) be its skewcharacteristic 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 \(e=uv\) and the summation is over all even cycles of G containing the edge e, and \(\operatorname{sgn}(C^{\sigma})\) denotes the sign of the even cycle \(C^{\sigma}\).
Lemma 2.4
([12], A part of Theorem 2.5)
Let \(G^{\sigma}\) be an oriented graph and \(\phi(G^{\sigma}, \lambda )\) be its skewcharacteristic polynomial. Then
where \(\frac{d}{d\lambda }\phi(G^{\sigma}, \lambda )\) denotes the derivative of \(\phi(G^{\sigma}, \lambda )\).
Some earlier results on the oriented graphs whose skewspectral radii do not exceed 2
Before proving the main theorems, we introduce some earlier results. By Lemmas 2.2 and 2.3 or the papers [10, 11], for a given graph G containing a cycle \(C_{m}\), we know that the skewspectral radius of \(G^{\sigma}\) is independent of its orientation if m is odd. Therefore we will briefly write G⃗ 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 \(\rho_{s}(G^{\sigma_{1}})\neq\rho_{s}(G^{\sigma_{2}})\). 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.
Firstly, we give a class of oriented graphs whose skewspectral radii do not exceed 2.
Denote by \(P_{l_{1},l_{2},\ldots,l_{k}}\) a pathlike graph, which is defined as follows: we first draw k (≥2) paths \(P_{l_{1}}, P_{l_{2}}, \ldots, P_{l_{k}}\) of orders \(l_{1},l_{2},\ldots,l_{k}\) 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 so that the four newly added edges form a cycle \(C_{4}\), where \(l_{1},l_{k}\ge0\) and \(l_{i}\ge1\) for \(i=2,3,\ldots,k1\). Then \(P_{l_{1},l_{2},\ldots,l_{k}}\) contains \(\sum_{i=1}^{k}l_{i}+2k2\) vertices. Notice that if \(l_{i}=1\) (\(i=2,3,\ldots,k1\)), the two end vertices of the path \(P_{l_{i}}\) are referred to as overlap; if \(l_{1}=0\) (\(l_{k}=0\)), the left (right) of the graph \(P_{l_{1},l_{2},\ldots,l_{k}}\) 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 \(l_{1}, l_{2}\geq1\). Meanwhile, \(P_{l_{1},l_{2},l_{3}}\), \(P_{0,l_{1},l_{2},l_{3}}\), \(P_{0,l_{1},l_{2},l_{3},0}\) are all bicyclic graphs containing \(C_{4}\), where \(l_{1}, l_{2},l_{3}\geq1\).
Then we have the following.
Lemma 3.1
([16])
Let \(P_{l_{1},l_{2},\ldots ,l_{k}}\) (\(k\ge2\)) be a pathlike graph described as above. Then
Moreover, 2 is an eigenvalue of \(P^{}_{l_{1},l_{2},\ldots,l_{k}}\) with multiplicity \(k2\) and \(\phi(P^{}_{l_{1},l_{2}}; 2) =4\).
Now, 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 radii are bounded by 2 are completely determined as follows.
Lemma 3.2
All undirected graphs whose spectral radii do not exceed 2 are \(C_{m}\), \(P_{0,n4,0}\), \(T_{2,2,2}\), \(T_{1,3,3}\), \(T_{1,2,5}\) and their subgraphs, where \(m\ge3\) and \(n\ge5\).
Consequently, combining with Lemma 2.2, the skewspectral radius of each oriented graph whose underlying graph is described as Lemma 3.2, regardless of the orientation of the oriented cycle \(C^{\sigma}_{m}\), does not exceed 2.
Let \(C_{m}=v_{1}v_{2}\cdots v_{m}v_{1}\) be a cycle on m vertices and \(P_{l_{1}},P_{l_{2}},\ldots,P_{l_{m}}\) be m paths with vertices \(l_{1},l_{2},\ldots,l_{m}\), respectively (perhaps some of them are empty). Denote by \(C_{m}^{l_{1},l_{2},\ldots,l_{m}}\) the unicyclic graph obtained from \(C_{m}\) by joining \(v_{i}\) to a pendent vertex of \(P_{l_{i}}\) for \(i=1,2,\ldots,m\). For convenience, suppose without loss of generality that \(l_{1}=\max\{l_{i}:i=1,2,\ldots,m\}\), \(l_{2}\ge l_{m}\) and write \(C_{m}^{l_{1},l_{2},\ldots,l_{j}}\) instead of the standard \(C_{m}^{l_{1},l_{2},\ldots,l_{j},0,\ldots,0}\) if \(l_{j+1}=l_{j+2}=\cdots=l_{m}=0\).
Denote by \(\theta_{a,b,c}\) the undirected bicyclic graph obtained from paths \(P_{a}\), \(P_{b}\), \(P_{c}\) by identifying the three initial vertices and terminal vertices of them, where \(\min\{a,b,c\}\ge2\) and at most one of them is 2, and by \(\infty_{m}^{p,q}\) (\(m\ge1\), \(p,q\ge3\)) the undirected bicyclic graph obtained from cycles \(C_{p}\) and \(C_{q}\) joined by a path \(P_{m}\). A bicyclic graph containing \(\theta_{a,b,c}\) (or \(\infty _{m}^{p,q}\)) is called of θtype (or ∞type). The set of bicyclic graphs of θtype (or ∞type) is denoted by \(\mathcal{G}_{\theta}\) (or \(\mathcal{G}_{\infty}\)). Furthermore, the subset of \(\mathcal{G}_{\theta}\) (or \(\mathcal{G}_{\infty}\)) containing \(\theta_{a,b,c}\) (or \(\infty_{m}^{p,q}\)) is denoted by \(\mathcal {G}_{\theta}(a,b,c)\) (or \(\mathcal{G}_{\infty}(p,q;m)\)).
On \(C_{4}\)free oriented graphs, we have the following.
Lemma 3.3
([16])
Let \(G^{\sigma}\) be an oriented graph and \(\rho_{s}(G^{\sigma})\le2\). Suppose that G is \(C_{4}\)free, then \(G^{\sigma}\) is one of the graphs \(C^{\sigma}_{m}\), \(P_{0,n4,0} \) (\(n\geq5\)), \(\vec{C}_{3}^{2}\), \((C_{6}^{2,0,0,2})^{}\), \((C_{6}^{1,0,1,0,1})^{}\), \((C_{8}^{1,0,0,0,1})^{}\), \(\theta^{\sigma}_{3,5,5}\) or their induced oriented subgraphs, where the orientation of \(C^{\sigma}_{m}\) is arbitrary. For induced even cycles \(C^{\sigma}_{6}\), \(C^{\sigma}_{8}\) of \(\theta^{\sigma}_{3,5,5}\), they satisfy \(\operatorname{sgn}(C^{\sigma}_{6})=1\) and \(\operatorname{sgn}(C^{\sigma}_{8})=1\).
A connected graph in which the number of edges equals the number of vertices is called a unicyclic graph. For convenience, we write

\(\mathcal{U}(m)=\{GG\mbox{ is a unicyclic graph containing the cycle }C_{m}\}\).
On oriented unicyclic graphs, we have the following.
Lemma 3.4
([17])
Let \(G^{\sigma}\) be an oriented unicyclic graph and \(\rho_{s}(G^{\sigma})\le2\). Then \(G^{\sigma}\) is one of the graphs \(C^{\sigma}_{m}\), \(\vec{C}_{3}^{2}\), \((C_{4}^{4,1})^{}\), \((C_{4}^{3,1,1})^{}\), \((C_{4}^{2,1,2,1})^{}\), \((C_{4}^{2,2})^{}\), \((C_{6}^{2,0,0,2})^{}\), \((C_{6}^{1,0,1,0,1})^{}\), \((C_{8}^{1,0,0,0,1})^{}\) and \(P^{}_{0,l_{1},l_{2},0}\) or their induced oriented unicyclic subgraphs, where the orientation of \(C^{\sigma}_{m}\) is arbitrary.
The oriented bicyclic graphs whose skewspectral radii do not exceed 2
In this section, we determine all the oriented bicyclic graphs whose skewspectral radii do not exceed 2. Let

\(\mathcal{B}_{2}=\{G^{\sigma}G\mbox{ is a bicyclic graph and }\rho_{s}(G^{\sigma})\le2\}\),

\(\mathcal{B}_{21}=\{G^{\sigma}G\mbox{ is a }C_{4}\mbox{free bicyclic graph and } \rho_{s}(G^{\sigma})\le2\}\),

\(\mathcal{B}_{22}=\{G^{\sigma}G\mbox{ is a bicyclic graph containing }C_{4} \mbox{ and }\rho_{s}(G^{\sigma})\le2\}\).
The main purpose of this paper is to determine the set \(\mathcal{B}_{2}\). Obviously, \(\mathcal{B}_{2}= \mathcal{B}_{21} \cup\mathcal{B}_{22}\). By Lemma 3.3, we immediately have the following result on the set \(\mathcal{B}_{21}\).
Theorem 4.1
\(\mathcal{B}_{21} = \{ \theta^{\sigma}_{3,5,5} \}\), where \(\operatorname{sgn}(C^{\sigma}_{6})=1\) and \(\operatorname{sgn}(C^{\sigma}_{8})=1\) for induced even cycles \(C^{\sigma}_{6}\), \(C^{\sigma}_{8}\) of \(\theta^{\sigma}_{3,5,5}\).
Now, we will determine the set \(\mathcal{B}_{22}\). Let

\(\mathcal{B}^{\infty}_{22}=\{G^{\sigma}G^{\sigma}\in\mathcal{B}_{22}\mbox{ and contain }\infty_{m}^{4,q}\ (m\ge1,q\ge3)\}\),

\(\mathcal{B}^{\theta}_{22}=\{G^{\sigma}G^{\sigma}\in\mathcal{B}_{22}\mbox{ and contain }\theta_{a,b,c}\ (c\geq b\geq a\geq2)\}\).
Obviously, \(\mathcal{B}_{22}= \mathcal{B}^{\infty}_{22} \cup\mathcal {B}^{\theta}_{22}\). On the set \(\mathcal{B}^{\infty}_{22}\), we have the following.
Theorem 4.2
\(\mathcal{B}^{\infty}_{22}=\{G^{\sigma}G^{\sigma}\textit{ is }P^{}_{0,l_{1},l_{2},l_{3},0}\ (l_{i}\geq1, i=1,2,3)\textit{ or its induced oriented bicyclic subgraphs}\}\).
Proof
We will complete the proof by considering following cases.
Case 1. \(q=3\).
We consider the graph \(\infty_{m}^{4,3} \) (\(m\ge1\)). Obviously, \(\infty _{m}^{4,3} \) (\(m\ge1\)) contains a unicyclic graph \(U_{1}\in\mathcal {U}(3)\), but \(U_{1}\) is not \(C_{3}^{2}\) or its induced unicyclic graphs. By Lemma 3.4, we know \(\rho_{s}((\infty_{m}^{4,3})^{\sigma})> 2\). Thus \(\rho _{s}(G^{\sigma})> 2\) for \(G\in\mathcal{G}_{\infty}(4,3;m)\).
Case 2. \(q\geq5\).
If \(q\neq6, 8\), then there exists a unicyclic graph \(U_{2} \in\mathcal {U}(q)  C_{q}\) in \(\infty_{m}^{4,q} \) (\(m\ge1\)). If \(q=6\), then there exists a unicyclic graph \(U_{3}\in\mathcal{U}(6)\) in \(\infty_{m}^{4,6}\) (\(m\ge1\)), but \(U_{3}\) is not any of the graphs \(C_{6}^{2,0,0,2}\), \(C_{6}^{1,0,1,0,1}\) or their induced unicyclic graphs. If \(q=8\), then there exists a unicyclic graph \(U_{4}\in\mathcal{U}(8)\) in \(\infty_{m}^{4,8} \) (\(m\ge1\)) but \(U_{4}\) is not \(C_{8}^{1,0,0,0,1}\) or its induced unicyclic graphs. So, by Lemma 3.4, we also have \(\rho_{s}((\infty_{m}^{4,q})^{\sigma})> 2 \) (\(q\geq5\)). Therefore \(\rho_{s}(G^{\sigma})> 2\) for \(G\in\mathcal{G}_{\infty}(4,q;m)\) (\(q\geq5\)).
Case 3. \(q=4\).
The sign of each induced cycle \(C^{\sigma}_{4}\) of \(G^{\sigma}\) must be negative. Furthermore, \(G^{\sigma}\) must be \(P^{}_{0,l_{1},l_{2},l_{3},0}\) or its induced oriented bicyclic subgraphs. Otherwise, G has an induced tree T such that T contains a proper subgraph \(P_{0,l,0}\). Hence \(\rho_{s}(G^{\sigma})> \rho_{s}(P_{0,l,0})= 2\). □
The rest of this manuscript is to determine the set \(\mathcal{B}^{\theta}_{22}\). Firstly, we have the following.
Lemma 4.1
If all the three cycles of \(\theta^{\sigma}_{a,b,c}\) are even, then there exists a cycle such that its sign is positive.
Proof
Suppose without loss of generality that \(\operatorname{sgn}(\theta^{\sigma}_{a,b})=\operatorname{sgn}(\theta^{\sigma}_{a,c})=1\), we will show \(\operatorname{sgn}(\theta^{\sigma}_{b,c})=1\).
Since
Thus
So
On the other hand,
But we know \(a + b\) is even. Then
□
Let \(G^{\sigma}\) be an oriented graph with the property
The property (4.1) is hereditary because, as a direct consequence of Lemma 2.1, for any induced subgraph \(H \subset G\), \(H^{\sigma}\) also satisfies (4.1). The inheritance (hereditary) of property (4.1) implies that there are minimal connected oriented graphs that do not obey (4.1); such graphs are called forbidden oriented subgraphs.
Let \(v_{1}v_{2}\cdots v_{b}v_{b+1}\cdots v_{b+c2}v_{1} \) (\(c\geq b\geq3\)) be the longest cycle in \(\theta_{2,b,c}\) and \(P_{b}= v_{1}v_{2}\cdots v_{b}\), \(P_{c}= v_{b}v_{b+1}\cdots v_{b+c2}v_{1}\). Denoted by \(\theta_{2,b,c}^{l_{1},l_{2},\ldots, l_{b}l_{b+1}\cdots l_{b+c2}}\) is the bicyclic graph obtained from \(\theta_{2,b,c}\) by joining its vertex \(v_{i}\) to a pendent vertex of \(P_{l_{i}}\) with vertices \(l_{i}\) (\(i=1,2,\ldots ,b+c2\)), where \(l_{i}\geq0\), \(l_{1}\geq l_{b}\). Moreover, if \(b=c\), we will suppose \(l_{2}\geq l_{b+c2}\).
Lemma 4.2
Let \(F_{1}= \theta^{1,0,0,0}_{2,3,3}\), \(F_{2}= \theta ^{0,1,0,0}_{2,3,3}\), \(F_{3}= \theta^{1,0,0,0,0}_{2,3,4}\), \(F_{4}= \theta ^{0,1,0,0,0}_{2,3,4}\), \(F_{5}= \theta^{0,0,0,1,0}_{2,3,4}\), \(F_{6}= \theta ^{2,0,0,0,0,0}_{2,4,4}\), \(F_{7}= \theta^{0,2,0,0,2,0}_{2,4,4}\) and \(F_{8}= \theta^{0,2,0,0,0,0,0,0}_{2,4,6}\). Then \(F^{\sigma}_{i} \) (\(i=1, 2, \ldots , 8\)) are forbidden oriented subgraphs on the property (4.1), where each subgraph \(C^{\sigma}_{4}\) of \(F^{\sigma}_{i} \) (\(i=1, 2, \ldots, 8\)) is negative. Also the subgraph \(C^{\sigma}_{6}\) of \(F^{\sigma}_{8}\) is negative.
Proof
By Lemma 2.3, it is not difficult to know that
Thus, by Lemma 3.1, \(\phi(\theta^{}_{2,3,3}; 2) =0\). And then \(\rho _{s}(\theta^{}_{2,3,3})=2\). Moreover, we have
Then \(\phi(F^{\sigma}_{1}; 2) =4\), and thus \(\rho_{s}(F^{\sigma}_{1}) > 2\). Also, we have
Then \(\phi(F^{\sigma}_{2}; 2) =2\), and therefore \(\rho_{s}(F^{\sigma}_{2}) > 2\).
Similarly, we have \(\rho_{s}(\theta^{}_{2,3,4})= \rho_{s}((\theta ^{1,0,0,0,0,0}_{2,4,4})^{\sigma})= \rho_{s}((\theta ^{0,2,0,0,1,0}_{2,4,4})^{\sigma})= \rho_{s}((\theta ^{0,1,0,0,0,0,0,0}_{2,4,6})^{\sigma})= 2\) and \(\rho_{s}(F^{\sigma}_{i})>2 \) (\(i=3, 4, \ldots, 8\)). □
In order to determine the set \(\mathcal{B}^{\theta}_{22}\), we will first consider the oriented graphs containing \(\theta^{\sigma}_{2,4,4}\) or \(\theta^{\sigma}_{3,3,c} \) (\(c \geq3\)). Combining with Lemma 4.2 and the fact that \(\rho_{s}(G^{\sigma})> 2\) if the oriented tree \(G^{\sigma}\) contains an arbitrary tree described in Lemma 3.2 as a proper subgraph, we have the following result.
Lemma 4.3
Let \(G \in\mathcal{G}_{\theta}(2,4,4)\) and \(G_{1}= \theta_{2,4,4}^{1,0,3,0,0,0}\), \(G_{2}= \theta_{2,4,4}^{1,0,1,0,1,0}\), \(G_{3}= \theta_{2,4,4}^{0,2,0,0,1,0}\) and \(G_{4}= \theta _{2,4,4}^{0,1,1,0,1,0}\). If \(\rho_{s}(G^{\sigma})\leq2\), then \(G^{\sigma}\) is one of the graphs \(G^{\sigma}_{i} \) (\(i=1,2,3,4\)) or their induced oriented bicyclic subgraphs, where each subgraph \(C^{\sigma}_{4}\) of \(G^{\sigma}_{i} \) (\(i=1,2,3,4\)) is negative.
Proof
Obviously, the sign of \(C^{\sigma}_{6}\) in \(\theta^{\sigma}_{2,4,4}\) must be positive. If \(G\neq\theta^{l_{1}, l_{2}, \ldots, l_{6}}_{2,4,4}\), we know \(\rho_{s}(G^{\sigma})> 2\) by Lemma 3.4.
Now, let \(G= \theta^{l_{1}, l_{2}, \ldots, l_{6}}_{2,4,4}\). By Lemma 4.2, we know \(l_{1} \leq1\).
Claim 1. If \(l_{1} =1\), then \(G^{\sigma}\) is one of the graphs \(G^{\sigma}_{1}\), \(G^{\sigma}_{2}\) or their induced oriented bicyclic subgraphs, where each subgraph \(C^{\sigma}_{4}\) of \(G^{\sigma}_{1}\) and \(G^{\sigma}_{2}\) is negative.
Obviously, we have \(l_{2}= l_{4} = l_{6} = 0\). Otherwise, \(\rho_{s}(G^{\sigma})> 2\) by Lemma 3.4. Suppose \(l_{3} \geq l_{5}\). Then \(l_{3} \leq3\). Otherwise \(\rho_{s}(G^{\sigma}) \geq\rho(T_{1,2,6})> 2\). Now, we consider the following two cases.
Case 1. \(l_{3} = 2\mbox{ or }3\).
Then \(l_{5}=0\). Otherwise \(\rho_{s}(G^{\sigma}) \geq\rho(T_{1,3,4})> 2\).
It is not difficult to know that \(\phi(G^{\sigma}_{1}; 2) = 0\). Thus \(\rho _{s}(G^{\sigma}_{1}) = 2\). Therefore \(G^{\sigma}\) is \(G^{\sigma}_{1}\) or its induced oriented bicyclic subgraphs in this case.
Case 2. \(l_{3}=1\).
We can obtain that \(\rho_{s}(G^{\sigma}_{2}) = 2\). Therefore \(G^{\sigma}\) is \(G^{\sigma}_{2}\) or its induced oriented bicyclic subgraphs in this case.
Claim 2. If \(l_{1}= 0\). Then \(G^{\sigma}\) is one of the graphs \((\theta_{2,4,4}^{0,3,0,0,0,0})^{\sigma}\), \((\theta _{2,4,4}^{0,1,0,0,0,1})^{\sigma}\), \(G^{\sigma}_{3}\), \(G^{\sigma}_{4}\) or their induced oriented bicyclic subgraphs, where each subgraph \(C^{\sigma}_{4}\) of the mentioned oriented graphs is negative.
Firstly, we can suppose \(l_{2}= \max \{l_{2}, l_{3}, l_{5}, l_{6}\}\geq1\). It is easy to see that \(l_{2} \leq3\), otherwise \(\rho_{s}(G^{\sigma}) \geq\rho (T_{1,3,4})> 2\) by Lemma 3.2. Now, we consider the following three cases.
Case 1. \(l_{2} = 3\).
Then \(l_{3}=l_{5}=l_{6}=0\). Otherwise \(\rho_{s}(G^{\sigma}) > 2\) by Lemma 3.2. On the other hand, \(\rho_{s}((\theta_{2,4,4}^{0,3,0,0,0,0})^{\sigma}) = 2\). Therefore \(G^{\sigma}\) is \((\theta_{2,4,4}^{0,3,0,0,0,0})^{\sigma}\) or its induced oriented bicyclic subgraphs in this case. Obviously, \(\theta _{2,4,4}^{0,3,0,0,0,0}\) is an induced subgraph of \(G_{1}\).
Case 2. \(l_{2}=2\).
Then \(l_{3}=l_{6}=0\) and \(l_{5} \leq2\) by Lemma 3.2. Moreover, by Lemma 4.2, we know \(l_{5} \leq1\). Since \(\rho_{s}(G^{\sigma}_{3}) = 2\). Therefore \(G^{\sigma}\) is \(G^{\sigma}_{3}\) or its induced oriented bicyclic subgraphs in this case.
Case 3. \(l_{2}=1\).
Subcase 3.1. \(l_{6}=1\).
Then \(l_{3}=l_{5}=0\). Since \(\rho_{s}((\theta_{2,4,4}^{0,1,0,0,0,1})^{\sigma}) = 2\). Therefore \(G^{\sigma}\) is \((\theta_{2,4,4}^{0,1,0,0,0,1})^{\sigma}\) or its induced oriented bicyclic subgraphs in this case. Obviously, \(\theta _{2,4,4}^{0,1,0,0,0,1}\) is an induced subgraph of \(G_{2}\).
Subcase 3.2. \(l_{6}=0\).
Then \(l_{3}\leq1\) and \(l_{5}\leq1\). But we have \(\rho_{s}(G^{\sigma}_{4}) = 2\). Therefore \(G^{\sigma}\) is \(G^{\sigma}_{4}\) or its induced oriented bicyclic subgraphs in this case.
Hence, we have completed the proof of this theorem. □
Now, we consider the oriented bicyclic graphs containing \(\theta^{\sigma}_{3,3,c} \) (\(c\geq3\)). For the graph \(\theta_{a,b,c}= \theta _{3,3,c}\), we suppose \(P_{a}=u_{1}u_{2}u_{3}\), \(P_{b}=v_{1}v_{2}v_{3}\), \(P_{c}=w_{1}w_{2}\cdots w_{c}\) and \(u_{1}=v_{1}=w_{1}\), \(u_{3}=v_{3}=w_{c}\). Also, in \(\theta^{\sigma}_{3,3,c}\), we suppose that the cycle \(C^{\sigma}_{4}\) is negative if c is even and only the cycle \(v_{1}v_{2}v_{3}w_{c1}\cdots w_{2}v_{1}\) is positive if c is odd. Furthermore, let \(T_{1,1,l} v=P_{1}\cup P_{1}\cup P_{l}\) and \(P_{l} = x_{1}x_{2}\cdots x_{l} \) (\(l\geq0\)). Denoted by \(G_{5}\) is the graph obtained from \(\theta_{3,3,3}\) by identifying \(u_{2}\) with \(x_{l}\) (or v if \(l=0\)) of \(T_{1,1,l}\). Suppose that \(G_{6}\) is the graph obtained from \(\theta_{3,3,5}\) by joining \(u_{2}\) to a pendent vertex of \(P_{2}\), and \(G_{7}\) is the graph obtained from \(\theta_{3,3,7}\) by joining \(u_{2}\) to an isolated vertex.
Lemma 4.4
Let \(G \in\mathcal{G}_{\theta}(3,3,c) \) (\(c\geq3\)) and \(\rho_{s}(G^{\sigma})\leq2\). Then \(G^{\sigma}\) is one of the graphs \(\theta^{\sigma}_{3,3,c}\) (\(c\geq4\), \(c\neq5,7\)), \(G^{\sigma}_{i} \) (\(i=5,6,7\)) or the induced oriented bicyclic subgraphs of \(G^{\sigma}_{i} \) (\(i=5,6,7\)), where the cycle \(C^{\sigma}_{4}\) is negative if c is even and only one cycle \(C_{c+1}\) in each \(\theta^{\sigma}_{3,3,c}\) is positive if c is odd.
Proof
We will complete the proof by proving the following five claims.
Claim 1. \(\rho_{s}(\theta^{\sigma}_{3,3,c}) = 2 \) (\(c\geq3\)).
By Lemma 2.3, we have
Thus \(\phi(\theta^{\sigma}_{3,3,c}; 2) =0\) because of \(\phi (P^{}_{l_{1},l_{2}}; 2) = 4 \) (\(l_{1}, l_{2}\geq0\)). And then \(\rho_{s}(\theta ^{\sigma}_{3,3,c}) = 2 \) (\(c\geq3\)).
Claim 2. If \(c\neq3,5,7\), then \(G^{\sigma}\) is one of the graphs \(\theta^{\sigma}_{3,3,c} \) (\(c\geq3\), \(c\neq3,5,7\)).
Suppose that H is the graph obtained from \(\theta_{3,3,c}\) by joining its one vertex to a pendent vertex. Then we know \(H^{\sigma}\) has an induced subgraph \((C^{1}_{k})^{\sigma}\), where \(k=5, 7\) or \(k\geq9\). Thus \(\rho_{s}(H^{\sigma})> 2\) and the result holds.
Claim 3. If \(c=3\), then \(G^{\sigma}\) is \(G^{\sigma}_{5}\) or its induced oriented bicyclic subgraphs.
If \(c=3\), then \(G^{\sigma}\) must be some oriented graph by joining a tree T to the vertex \(u_{2}\) of \(\theta_{3,3,3}\). By Lemma 3.4, we know \(G^{\sigma}\) can only be \(G^{\sigma}_{5}\) or its induced oriented bicyclic subgraphs. Now we will show \(\rho_{s}(G^{\sigma}_{5}) = 2\).
Let \(\theta^{l}_{3,3,3}\) be the graph obtained from \(G_{5}\) by deleting its two pendent vertices. We show by induction on l that \(\rho _{s}((\theta^{l}_{3,3,3})^{\sigma})=2\). If \(l=0\), then \(\rho_{s}((\theta ^{o}_{3,3,3})^{\sigma})=\rho_{s}(\theta^{\sigma}_{3,3,3})=2\). Suppose now that \(l\geq0\) and the result is true for the order no more than l. Then we have
So \(\phi((\theta^{l+1}_{3,3,3})^{\sigma}; 2)= 0\) if \(l=0\) because of \((\theta^{1}_{3,3,3})^{\sigma}=C^{+}_{4}\). Therefore \(\rho_{s}((\theta ^{l}_{3,3,3})^{\sigma})=2\).
Now, we will prove \(\rho_{s}(G^{\sigma}_{5})=2\). Firstly, we have
Then 2 is an eigenvalue of \(G_{5}\).
Also, by Lemma 2.4, we have
Since \(\rho_{s}(G^{\sigma}_{5}v)=2\) for each \(v\in V(G_{5})\), we know 2 is an eigenvalue of \(G_{5}\) with multiplicity 2. But \(\lambda _{3}(G_{5})< 2\), so we can confirm the result holds.
Claim 4. If \(C=5\), then \(G^{\sigma}\) is \(G^{\sigma}_{6}\) or its induced oriented bicyclic subgraphs.
In this case, \(G^{\sigma}\) must be some oriented graph by joining a path to the vertex \(u_{2}\) of \(\theta^{\sigma}_{3,3,5}\). By Lemma 3.4, we know \(G^{\sigma}\) can only be \(G^{\sigma}_{6}\) or its induced oriented bicyclic subgraphs. To confirm the result holds, we only need to show \(\rho_{s}(G^{\sigma}_{6}) = 2\). In fact, similar to the proof of Claim 3, we can show 2 is an eigenvalue of \(G_{6}\) with multiplicity 2. And thus \(\rho_{s}(G^{\sigma}_{6}) = 2\).
Claim 5. If \(c=7\), then \(G^{\sigma}\) is \(G^{\sigma}_{7}\) or its induced oriented bicyclic subgraphs.
Similarly, we know the result holds. □
Theorem 4.3
\(\mathcal{B}^{\theta}_{22}=\{ \theta^{\sigma}_{2,3,3}, \theta^{\sigma}_{2,3,4}, (\theta ^{0,1,0,0,0,0,0,0}_{2,4,6})^{\sigma}, \theta^{\sigma}_{3,3,c} \ (c\geq 4, c\neq5,7), G^{\sigma}_{i} \ (i=1, 2,\ldots, 7) \textit{ or their induced oriented bicyclic subgraphs}\}\), where the cycle \(C^{\sigma}_{4}\) is negative if c is even and only one cycle \(C_{c+1}\) in each \(\theta^{\sigma}_{3,3,c}\) is positive if c is odd.
Proof
Firstly, since \(2\leq a\leq b\leq c\) and \(G^{\sigma}\) containing \(C_{4}\), we know \(2\leq a\leq3\), \(3\leq b\leq4\). Furthermore, the radius of each graph mentioned in this theorem is 2. The discussion is divided into three parts according to different cases of a and b.
Case 1. \(a=2\), \(b=3\).
Subcase 1.1. \(c=3\) or \(c=4\).
By Lemma 4.2, we know \(G^{\sigma}\) must be \(\theta^{\sigma}_{2,3,3}\) or \(\theta^{\sigma}_{2,3,4}\).
Subcase 1.2. \(c\geq5\).
Then \(\rho_{s}(G^{\sigma}) \geq\rho_{s}((C_{3}^{1,1})^{\sigma})> 2\).
Case 2. \(a=2\), \(b=4\).
By Lemma 3.4 and Lemma 4.2, we know \(c=4\) or \(c=6\).
Subcase 2.1. \(c=4\).
By Lemma 4.3, \(G^{\sigma}\) must be one of the graphs \(G^{\sigma}_{i}\) (\(i=1,2,3,4\)) or their induced oriented bicyclic subgraphs.
Subcase 2.2. \(c=6\).
By Lemma 4.2, \(G^{\sigma}\) must be \((\theta ^{0,1,0,0,0,0,0,0}_{2,4,6})^{\sigma}\) or its induced oriented bicyclic subgraph.
Case 3. \(a=3\).
Then \(b=3\), otherwise \(G^{\sigma}\) does not contain \(C_{4}\). By Lemma 4.4, \(G^{\sigma}\) must be \(\theta^{\sigma}_{3,3,c} \) (\(c\geq3\), \(c\neq 3,5,7\)), \(G^{\sigma}_{i} \) (\(i=5,6,7\)) or their induced oriented bicyclic subgraphs. □
Combining with Theorem 4.1, Theorem 4.2 and Theorem 4.3, we have the following main result.
Theorem 4.4
\(\mathcal{B}_{2} = \{ \theta^{\sigma}_{3,5,5}, P^{}_{0,l_{1},l_{2},l_{3},0}, \theta^{\sigma}_{2,3,3}, \theta^{\sigma}_{2,3,4}, (\theta^{0,1,0,0,0,0,0,0}_{2,4,6})^{\sigma}, \theta^{\sigma}_{3,3,c}\ (c\geq4, c\neq5,7), G^{\sigma}_{i} \ (i=1, 2,\ldots, 7)\textit{ or their induced oriented bicyclic subgraphs}\}\), where the cycle \(C^{\sigma}_{4}\) is negative if c is even and only one cycle \(C_{c+1}\) in each \(\theta^{\sigma}_{3,3,c}\) is positive if c is odd.
By this theorem, we have the following.
Theorem 4.5
Let \(G^{\sigma}\) be an oriented bicyclic graph and \(\rho_{s}(G^{\sigma})< 2\). Then \(G^{\sigma}\) is one of the graphs \((\theta _{2,4,4}^{0,1,0,0,1,0})^{\sigma}\), \((\theta_{2,4,4}^{0,2,0,0,0,0})^{\sigma}\), \(\theta^{\sigma}_{2,4,6}\) or their induced oriented bicyclic subgraphs.
Proof
Of course, the radius of each graph mentioned in this theorem is less than 2. Obviously, by Theorem 4.4, G must contain \(\theta_{2,4,4}\) or \(\theta_{2,4,6}\) if \(\rho_{s}(G^{\sigma})< 2\). Firstly, by taking some direct calculations, we know \(\phi(\theta^{\sigma}_{2,4,6}; 2)=1\), and then \(\rho_{s}(\theta^{\sigma}_{2,4,6})< 2\).
Now suppose that G contains \(\theta_{2,4,4}\). By taking some direct calculations, we know \(\rho_{s}((\theta_{2,4,4}^{1,0,0,0,0,0})^{\sigma})= 2\). Thus \(l_{1}=l_{4}=0\). Similarly, we have \(\rho_{s}((\theta _{2,4,4}^{0,3,0,0,0,0})^{\sigma})= 2\). Thus \(l_{2}= \max\{l_{2},l_{3},l_{5},l_{6}\} \leq2\).
Case 1. \(l_{2}=2\).
Since \(\rho_{s}((\theta_{2,4,4}^{0,2,1,0,0,0})^{\sigma})> 2\), \(\rho _{s}((\theta_{2,4,4}^{0,2,0,0,0,1})^{\sigma})> 2\) and \(\rho_{s}((\theta_{2,4,4}^{0,2,0,0,1,0})^{\sigma})= 2\). Thus \(G^{\sigma}\) must be \((\theta_{2,4,4}^{0,2,0,0,0,0})^{\sigma}\) or its induced oriented bicyclic subgraphs in this case.
Case 2. \(l_{2}=1\).
Since \(\rho_{s}((\theta_{2,4,4}^{0,1,0,0,0,1})^{\sigma})= 2\) and \(\rho _{s}((\theta_{2,4,4}^{0,1,1,0,0,0})^{\sigma})= 2\). Thus \(G^{\sigma}\) must be \((\theta_{2,4,4}^{0,1,0,0,1,0})^{\sigma}\) or its induced oriented bicyclic subgraphs in this case. □
By this theorem, we immediately obtain the following sharp lower bound on the skewspectral radii of oriented bicyclic graphs.
Corollary 4.1
Let \(G^{\sigma}\) be an oriented bicyclic graph on order n and \(n\geq9\). Then \(\rho_{s}(G^{\sigma})\geq2\).
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This work was supported by the National Natural Science Foundation of China (No. 11171373).
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Xu, GH., Gong, SC. The oriented bicyclic graphs whose skewspectral radii do not exceed 2. J Inequal Appl 2015, 326 (2015). https://doi.org/10.1186/s1366001508484
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DOI: https://doi.org/10.1186/s1366001508484
MSC
 05C20
 05C50
 15A18
 15C35
Keywords
 oriented graph
 skewadjacency matrix
 skewspectral radius