The ﬁrst three largest values of the spectral norm of oriented bicyclic graphs

Let D be a digraph with n vertices and let σ 1 ( D ) , σ 2 ( D ) , . . . , σ n ( D ) be the singular values of the adjacency matrix of D , where σ 1 ( D ) ≥ σ 2 ( D ) ≥ · · · ≥ σ n ( D ) . The spectral norm of D is σ 1 ( D ) . In this paper, we determine the orientations of graphs with the ﬁrst three largest values of the spectral norm over the family of all orientations of bicyclic graphs with at least 12 vertices.


Introduction
We consider digraphs without loops or multiple arcs. Let D be a digraph with vertex set V (D) and arc set E(D). The notations and terminologies used but not defined here can be found in [4,5]. Denote by uv the arc from vertex u to vertex v (i.e. the arc with tail u and head v). The out-degree (in-degree, respectively) of a vertex u of D, denoted by d + D (u) (d − D (u), respectively), is the number of arcs of the form uv (vu, respectively) in D. A vertex u with d + D (u) = 0 (d − D (u) = 0, respectively) is called a sink (source, respectively) of D. The transpose D of a digraph D is obtained from D by reversing all arcs.
The adjacency matrix of an n-vertex digraph D is the n × n matrix A(D) = (a uv ) u,v∈V (G) , where a uv = 1 if uv ∈ E(D) and 0 otherwise. Let λ 1 (D), λ 2 (D), . . . , λ n (D) denote the eigenvalues of A(D). Since A(D) is not necessarily symmetric, its eigenvalues are not necessarily real numbers. The spectral radius of D is defined as ρ(D) = max{|λ i | : i = 1, 2, . . . , n}.
We mention that a (simple undirected) graph G corresponds naturally to a digraph D(G) with the same vertex set such that if there is an edge connecting vertices u and v in G, then there are arcs uv and vu in D(G). The adjacency matrix of G is A(G) = A(D(G)). The spectral radius ρ(G) is the largest eigenvalue of its adjacency matrix.
For an n × n real matrix M , the singular values, σ 1 (M ) ≥ σ 2 (M ) ≥ · · · ≥ σ n (M ) of M are the nonnegative square roots of the eigenvalues of M T M or, equivalently, of M M T . The largest singular value, σ 1 (M ), is called the spectral norm of M . For a digraph D, the spectral norm σ 1 (D) is the spectral norm of A(D).
An orientation of a graph G is a digraph D obtained by choosing a direction for each edge of G. In this case, we say that D is an orientation of G and G is the underlying graph of D. A source-sink orientation (SS-orientation for short) of a graph G is an orientation such that each vertex is either a source or a sink. Monsalve and Rada [14] obtained that G has a SS-orientation if and only if G is bipartite.
A connected graph G is a bicyclic graph if |E(G)| = |V (G)| + 1. Let B 1 n (B 2 n , respectively) be the n-vertex bicyclic graph obtained by adding two adjacent (nonadjacent, respectively) edges to the star S n . Let D n,1 , D n,2 and D n,3 be the orientations of B 1 n as shown in Figure 1. Let D n,3 be the orientation of B 2 n as shown in Figure 1. Some extremal problems in spectral digraph theory have attracted a great deal of research; some specific results on extremal problems for digraphs can be found in [1][2][3]10,11,13]. Gregory and Kirkland [7] obtained lower and upper bounds on the spectral norm of a tournament and determined the tournament with maximum spectral norm. In other words, they found the orientation of K n attaining maximum spectral norm over the set of all orientations of K n . Hoppen, Monsalve and Trevisan [9] obtained the extremal values of the spectral norm over the set of oriented trees, oriented unicyclic graphs and connected digraphs with n vertices and n arcs. García, Monsalve and Rada [6] obtained lower bounds for the spectral norm of a digraph in terms of the structure of the digraph. In this paper, we found the orientations of bicyclic graphs attaining the first three largest values of the spectral norm over the family of all orientations of bicyclic graphs.  Note that if D is an n-vertex digraph with arcs such that exactly k vertices are neither sinks nor sources, then the D-stretched digraph D has n + k vertices and arcs. Moreover, all vertices of D are sinks or sources. Hence, D is a SS-orientation of a bipartite graph that will be denoted by H D . Obviously, H D is the underlying graph of the digraph D.

Preliminaries
Obviously, D doesn't depend on the order in which the vertices of D are stretched. In the sense of isomorphism, these maximal vertex-disjoint digraphs whose vertices are either sources or sinks of D may be viewed as maximal arc-disjoint subdigraphs whose vertices are either sources or sinks of D. Thus we call these maximal vertex-disjoint digraphs whose vertices are either sources or sinks of D the maximal SS-subdigraphs of D or D.   For integers a and b with b ≥ a, the tree obtained by adding an edge between the centers of two vertex-disjoint stars S a+1 and S b+1 is denoted by S n,a . Obviously, the star S n ∼ = S n,0 . Let S n,3 be the tree obtained by attaching two pendent edge to two pendent vertices of S n−2 . Lemma 2.3 (see [8]). Let T be a n-vertex tree and T ∼ = S n , S n,1 , S n,2 , S n,3 . Then ρ(T ) < ρ(S n,3 ) < ρ(S n,2 ) < ρ(S n,1 ) < ρ(S n ).
Let U n,1 be the unicyclic graph obtained from C 4 by attaching n − 4 pendent vertices to v 1 and U n,2 be the unicyclic graph obtained from C 4 by attaching n − 5 pendent vertices to v 1 and a pendent vertex to v 2 . Lemma 2.4 (see [12]). Let G be a n-vertex unicyclic bipartite graph different from U n,1 and U n,2 . Then Let B n be the graph obtained by joining n − 5 pendent vertices to a vertex of degree three of the complete bipartite graph K 2,3 . Lemma 2.5 (see [16]). Let G be a bicyclic bipartite graphs with n ≥ 5 vertices, then with equality if and only if G ∼ = B n . Lemma 2.6 (see [15]). Let G be a connected graph. If H is a proper subgraph of G, then ρ(H) < ρ(G). Proof. Note that U n+1,1 is the underlying graph of D n,1 and D n,1 . By Lemma 2.2, we have

Main results
Let D 1 , . . . , D k be maximal vertex-disjoint SS-subdigraphs of D and H i be the underlying graph of D i for 1 ≤ i ≤ k. Without loss of generality, we assume that By Theorem 2.1 and Lemma 2.2, we have σ 1 (D) = σ 1 (D 1 ) = ρ(H 1 ). Note that H 1 is a tree, a bipartite unicyclic graph or a bipartite bicyclic graph and ∆(H 1 ) ≤ n − 1.
Suppose that s = n + 1. Then by Lemma 2.4, ρ(H 1 ) ≤ ρ(U n+1,1 ). If the equality holds, then H 1 ∼ = U n+1,1 . Thus D = D(v) and D is the SS-orientation of U n+1,1 , where v is the unique vertex nither sink nor source. Therefore, D can be obtained from D by identifying a sink u 1 with a source u 2 . Thus D ∼ = D n,1 or D n,1 . Suppose now that H 1 is a bipartite bicyclic graph with s vertices. Then s ≤ n. By Lemma 2.5, for n ≥ 10.
Since the proof of the next result is similar to the proof of Theorem 3.2, we omit it.