On the least signless Laplacian eigenvalue of a non-bipartite connected graph with fixed maximum degree

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively.


Introduction
All graphs considered in this paper are finite, simple and undirected. Let G be a graph with vertex set V = V (G) = {v  , v  , . . . , v n } and edge set E = E(G). Write A(G) for the adjacency matrix of G and let D(G) be the diagonal matrix of the degrees of G. The matrix Q(G) = D(G)+A(G) is called the signless Laplacian matrix of G. As usual, let q  (G) ≥ q  (G) ≥ · · · ≥ q n (G) ≥  denote the eigenvalues of Q(G) and call them the signless Laplacian eigenvalues of G. Denote by κ(G) the least eigenvalue of G.
For a connected graph G, κ(G) =  if and only if G is bipartite. Desai and Rao [] suggest the use of κ(G) as a measure of non-bipartiteness of G. Fallat and Fan [] introduce two parameters reflecting the graph bipartiteness, and establish a relationship between κ(G) and the two parameters. de Lima, Nikiforov and Oliveira [] point out that κ(G) depends more on the distribution of the edges of a graph than on their number, so it may become a useful tool in extremal graph theory. For a connected non-bipartite graph G with given order, how small can κ(G) be? Cardoso et al. [] propose this problem and show that the minimum value of κ(G) is attained uniquely in the unicyclic graph obtained from the cycle C  by attaching a path at one of its end vertices. Motivated by this problem, a good deal of attention has been devoted to finding all graphs with the minimal least signless Laplacian eigenvalue among a given class of graphs. For related results, one may refer to [-].
A unicyclic graph is a connected graph with a unique cycle. Let = (G) be the maximum degree of a graph G. In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree and among all non-bipartite connected graphs of order n with maximum degree , respectively.
The rest of the paper is organized as follows. In Section , we recall some notions and lemmas used further, and prove three new lemmas. In Section , we prove two theorems which is our main result. In Section , we propose two problems for further research.

Preliminaries
Denote by C n the cycle on n vertices. Let Guv denote the graph which arises from G by deleting the edge uv ∈ E(G). Similarly, G + uv is the graph that arises from G by adding Let x ∈ R n be an arbitrary unit vector. One can find in [, ] that with equality if and only if x is an eigenvector corresponding to κ(G). Let G  and G  be two vertex-disjoint connected graphs, and let v i ∈ V (G i ) for i = , . Identifying the v  with v  and forming a new vertex u (see [] for details), the resulting graph is called coalescence of G  and G  , and denoted by

Lemma . ([]) Let G be a connected graph which contains a bipartite branch B with root u, and x be an eigenvector corresponding to κ(G).
( and with root u. Then |x(q)| < |x(p)| whenever p, q are vertices of T such that q lies on the unique path from u to p.

Lemma . ([]) Let G be a non-bipartite connected graph, and let x be an eigenvector corresponding to κ(G). Let T be a tree, which is a nonzero branch of G with respect to x
For k ≥ , let G denote the graph obtained from G by deleting the edge uv, inserting Then G is called a k-subdivision graph of G by k-subdividing the edge uv.

Lemma . Let G = G  (v) B(v) be a connected graph, where G  is a graph of order n, and B is a bipartite graph of order s. Then κ(G) ≤ κ(G  ). Moreover, if s > , G  is nonbipartite and there exists an eigenvector x corresponding to
. . , x(v n )) T be a unit eigenvector corresponding to κ(G  ). Then Without loss generality, we may assume v = v n . Let V (B) = {v n , v n+ , . . . , v n+s- }, and let (U, W ) be the two parts of the bipartite graph B, Clearly, if s > , G  is non-bipartite and x(v) = , we have y  > x  . This implies that κ(G) < κ(G  ).
Lemma . Let n ≥  and s ≥  be integer. G  and G  , shown in Figure , are two unicyclic graphs of order n. Then κ(G  ) < κ(G  ).
Proof Let κ = κ(G  ), and x = (x  , x  , . . . , x n ) T be a unit eigenvector corresponding to κ. Then κ = v i v j ∈E(G  ) (x i + x j )  and  < κ < . By Lemmas . and ., we have x n = . From the eigenvalue equation Q(G  )x = κx, we have x n- = κ  -κ  + κ  -κ +  x n , Let y = (y  , y  , . . . , y n ) T ∈ R n defined on V (G  ) satisfy that Let f (t) = t  -t  + t  -t  -t  + t  -t + . By a computation, f (t) =  has five real roots which are approximately equal to -., ., ., , ., respectively. By Lemma ., we have Note that G v  -· · ·v s+ is a t-subdividing graph of G  or G  (shown in Figure ). By Lemma ., we have By a computation, we have κ(G  ) ≈ . and κ(G  ) ≈ .. It follows that κ < .. Noting that f () = , we have f (κ) > . It follows that y  > x  . Combining the above arguments, we have This completes the proof.
Proof Let κ = κ(U * n (, n -)), and x = (x  , x  , . . . , x n ) T be a unit eigenvector corresponding to κ. By Corollary . of [], it is easy to see κ(G) < /. From the eigenvalue equation This implies that κ is the least root of the following equation: Similarly, we can see that κ(U n- n ()) is the least root of the following equation: Noting that g() = - <  and for  < x < /, we have g(κ) > , and so κ U n- n () < κ = κ U * n (, n -) .

Main results
Let U(n, ) be the set of non-bipartite unicyclic graphs of order n with maximum degree , and G(n, ) be the set of non-bipartite connected graphs of order n with maximum degree . In this section, we firstly determine the unicyclic graph whose signless Laplacian eigenvalue attains the minimum among all graphs in U(n, ).
Theorem . Let  ≤ ≤ n -. Among all graphs in U(n, ), U - n () is the unique graph whose signless Laplacian eigenvalue attains the minimum.
Proof Let G ∈ U(n, ), and C g = v  v  . . . v g v  be the unique cycle of G. Then g is odd, and G can be obtained by attaching trees T  , T  , . . . , T g to the vertices v  , v  , . . . , v g of C g , respectively, where T i contains the root vertex v i for i = , , . . . , g. |V (T i )| =  means V (T i ) = {v i }. Suppose that G has k pendant vertices. It is easy to see ≤ k + . Let x = (x  , x  , . . . , x n ) T be a unit eigenvector corresponding to κ(G).
Case . ≤ k + . By Lemma ., we have κ(U k n ()) ≤ κ(G) with equality if and only if G = U k n (). By Lemma ., we have κ(U - n ()) ≤ κ(U k n ()) with equality if and only if = k + . It follows that κ(U - n ()) ≤ κ(G) with equality if and only if G = U - n (). Case . = k + . Then G must be the graph obtained from the cycle C g with k pendant paths P i  , . . . , P i k attached at the same vertex v  of C g , and k ≥ .
() Characterize all extremal graphs whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite connected graphs with a given degree sequence.