New Bounds for the Signless Laplacian Spread

Let $G$ be a simple graph. The signless Laplacian spread of $G$ is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.


Introduction
In this paper we study an spectral invariant called signless Laplacian spread, defined as the difference between the maximum and minimum signless Laplacian eigenvalues. Some inequalities using the vertex bipartiteness value are explored and some relations between the mentioned parameter and the signless Laplacian spread are studied.
We deal with a simple graph G with vertex set V(G) of cardinality n and edge set E(G) of cardinality m; we call this an (n, m)-graph. An edge e ∈ E(G) with end vertices u and v is denoted by uv, and we say that u and v are neighbors. A vertex v is incident to an edge e if v ∈ e. N G (v) is the set of neighbors of the vertex v, and its cardinality is the degree of v, denoted by d(v). Sometimes, after a labeling of the vertices of G, a vertex v i is simply written i and an edge v i v j is written ij, and we write d i for d(v i ). The minimum and maximum vertex degree of G are denoted by δ(G) (or simply δ) and ∆(G) (or simply ∆), respectively. A q-regular graph G is a graph where every vertex has degree q. K n is the complete graph of order n, and P n (resp. C n ) is the path (resp. cycle) with n vertices. A bipartite graph with bipartition (X, Y ) is denoted by G(X, Y ) (so any edge has one end vertex in X and the other in Y ). K p,q is the complete bipartite graph with bipartition (X, Y ) with |X| = p, |Y | = q. A graph is called semi-regular bipartite if it is bipartite and the vertices belonging to the same part have equal degree. We denote by G ∪ H the vertex disjoint union of graphs G and H. A subgraph H of G is an induced subgraph whenever two vertices of V (H) are adjacent in H if and only if they are adjacent in G. If H is an induced subgraph of G, the graph G − H is the induced subgraph of G whose vertex set is V (G − H) = V(G) \ V(H). We only consider graphs without isolated vertices. Let d 1 , d 2 , . . . , d n be the vertex degrees of G. A G = (a ij ) denotes the adjacency matrix of G. The spectrum of A G is called the spectrum of G and its elements are called the eigenvalues of G. The vertex degree matrix D G is the n × n diagonal matrix of the vertex degrees d i of G. The signless Laplacian matrix of G (see e.g. [9]) is defined by So, if Q G = (q ij ), then q ij = 1 when ij ∈ E(G), q ii = d i , and the remaining entries are zero. The signless Laplacian matrix is nonnegative and symmetric. The signless Laplacian spectrum of G is the spectrum of Q G . Similarly, the matrix is the Laplacian matrix of G ( [9,16,17]). For all these matrices we may omit the subscript G if no misunderstanding should arise. For a real symmetric matrix W G , associated to a graph G, its spectrum (the multiset of the eigenvalues of W G ) is denoted by σ WG , and we let η i (W G ) denote the i'th largest eigenvalue of W G . The i'th largest eigenvalue of A G (L G , Q G , respectively) is denoted by λ i (G) (µ i (G), q i (G), respectively). Sometimes they are simply denoted by λ i (µ i , q i , respectively).
For a graph G, its line graph L G is the graph with vertex set E(G) and where two edges in E(G) are adjacent in L G whenever the corresponding edges in G have a common vertex.
Let I G be the vertex-edge incidence matrix of the (n, m)-graph G, defined as the n × m matrix whose (i, j)-entry is 1 if vertex v i is incident to the edge e j , and 0 otherwise. It is well known (see e.g. [5,9]) that where I m denotes the identity matrix of order m, so the matrices Q G and 2I m + A LG share the same nonzero eigenvalues. As a consequence, if G is an (n, m)-graph, then q i (G) = 2 + λ i (L G ) for i = 1, 2, . . . , k where k = min {m, n} and λ i (L G ) is the i'th largest eigenvalue of L G . Moreover, if m > n, then λ i (L G ) = −2, for m ≥ i ≥ n + 1 and if n > m, then q i = 0 for n ≥ i ≥ m + 1.
An orientation of a graph G is the directed graph obtained from G by replacing every edge uv by one of the pairs (arcs) (u, v) or (v, u). We let O(G) denote the set of all orientations of G. For G ′ ∈ O(G), its (0, ±1)-incidence matrix, denoted by K G ′ = (ρ ij ), i = 1, 2, . . . , n; j = 1, 2, . . . , m, is given by Then, whatever the orientation of the edges ( [19,28]), the matrix K G ′ satisfies the following identity By the relations (2) and (4) it is clear that the matrices Q G and L G are both positive semidefinite. Moreover, the all ones n-dimentional vector e is an eigenvector of the Laplacian matrix for the eigenvalue 0. Note: We treat vectors in R n as column vectors, but identify these with the corresponding n-tuples.

The spread of symmetric matrices
This section collects some general results that are known for the spread of a symmetric matrix.
Let ω i be the i'th largest eigenvalue of a symmetric matrix W . The spread of W is defined by There are several papers devoted to this parameter, see for instance [22,23,29,31]. For a real rectangular matrix W = (w ij ), let W F = ( ij w 2 ij ) 1/2 be the Frobenius norm of W . When W is square its trace will be denoted by tr W . In 1956, Mirsky proved the following inequality.
Theorem 1 ( [29]) Let W be an n × n normal matrix. Then with equality if and only if the eigenvalues ω 1 , ω 2 , . . . , ω n of W satisfy the following condition Among the results obtained for the spread of a symmetric matrix W = (w ij ) we mention the following lower bound obtained in [3].
be an n × n normal and symmetric matrix. Then Some other lower bounds for the spread of Hermitian matrices are found in [22], and in some cases these improve the lower bound in (6).

Theorem 3 ([22]) For any Hermitian matrix
where e ij = 2f ij if w ii = w jj and otherwise

Spreads associated with graphs
Let G be an (n, m)-graph. We now consider different notions of spread based on matrices associated with G.
As before A G is the adjacency matrix of G and we consider which is called the spread of G ( [15]). Let µ(G)= (µ 1 , µ 2 , . . . , µ n ) be the vector whose components are the Laplacian eigenvalues of G (ordered decreasingly, as usual). The Laplacian spread, denoted by s L (G), is defined ( [41]) by Note that µ n = 0. Let q(G) = (q 1 , q 2 , . . . , q n ) be the vector whose components are the signless Laplacian eigenvalues ordered decreasingly. The signless Laplacian spread, denoted by s Q (G), is defined ( [26], [32]) as Remark 4 Some basic properties of these notions are as follows: (i) Let G be a graph of order n with largest vertex degree ∆. From Theorem 2 one can easily see that s(G) ≥ 2 √ ∆. Moreover, if G = K 1,n−1 , equality holds.
(iii) From the relation Q G = 2A G + L G it follows that q 1 ≥ 2λ 1 as L G is positive semidefinite (and it is known that equality holds if an only if G is a regular graph), (see e.g [8,12]). Moreover, as λ 1 is the spectral radius of A G , 2λ 1 ≥ λ 1 − λ n = s(G) with equality if and only if G is a bipartite graph. Therefore s(G) ≤ q 1 with equality if and only if G is a regular, bipartite graph.
(iv) We recall the Weyl's inequalities for a particular case in what follows. Consider two n×n Hermitian matrices W and U with eigenvalues (ordered nonincreasingly) ω 1 , ω 2 , . . . , ω n and x 1 , x 2 , . . . , x n , respectively, and the Hermitian matrix T = W + U with eigenvalues τ 1 , τ 2 , . . . , τ n (ordered nonincreasingly). Then the following inequalities hold Thus ω n + x 1 ≤ τ 1 ≤ ω 1 + x 1 and ω n + x n ≤ τ n ≤ ω 1 + x n . Therefore which gives the following inequalities for the spread of these matrices (v) Let G be a graph with smallest and largest vertex degree δ and ∆, respectively. By the previous item, as The next inequality establishes a relation between the largest Laplacian eigenvalue and the largest signless Laplacian eigenvalue. Lemma 5 ([40]) Let G be a graph. Then Moreover if G is connected, then the equality holds if and only if G is a bipartite graph.
The second smallest Laplacian eigenvalue of a graph G is known as the algebraic connectivity ( [13]) of G and denoted by a(G). If G is a non-complete graph, then a(G) ≤ κ 0 (G), where κ 0 (G) is the vertex connectivity of G (that is, the minimum number of vertices whose removal yields a disconnected graph). Since κ 0 (G) ≤ δ(G), it follows that a(G) ≤ δ. The graphs for which the algebraic connectivity attains the vertex connectivity are characterized in [25]. One also has ( [2,39]) For a survey on algebraic connectivity, see [1]. Moreover, it is worth to conclude that the result in (7) together with the result in Remark 4 (v) imply that

Remark 6
If G is a connected (n, m)-graph such that m ≤ n − 1, then G does not have cycles and thus, it is bipartite. Therefore s Q (G) = q 1 = µ 1 . As in the literature there are many known lower and upper bounds for this eigenvalue, from now on we only treat the case m ≥ n.
Some results on s Q (G) can be found in [32,39]. Some of them are listed below.
Theorem 7 ( [32,42]) For any graph G with n vertices with equality if and only if G = P n or G = C n in case n odd.
and equality holds if and only if G = K n−1 ∪ K 1 .
where the upper bound holds with equality if and only if G is regular or semiregular bipartite.
The minimum number of vertices (resp., edges) whose deletion yields a bipartite graph from G is called the vertex bipartiteness (resp., edge bipartiteness) of G and it is denoted υ b (G) (resp., ǫ b (G)), see [11]. Let q n be the smallest eigenvalue of Q G . In [11], one established the inequalities In [42] some important relationships between ǫ b (G) and s Q (G) were found, and it was shown that if with equality if and only if G is one of the following graphs: K 1,3 , K 4 , two triangles connected by an edge, and C n with n even.

Lower bounds
We now present some new lower bounds for the signless Laplacian spread. The first results involve the vertex bipartiteness graph invariant.
Theorem 10 Let G be a connected graph with n vertices and vertex bipartite- If G is a connected bipartite graph, then equality holds in (9).
Proof. By Lemma 5 By using (8) the inequality in the statement follows. If G is a bipartite graph, then υ b (G) = 0 = q n and q 1 (G) = µ 1 (G) then the equality follows.

Corollary 11
Let G be a connected (n, m)-graph and vertex bipartiteness υ b (G).
Equality holds here if G is a regular bipartite graph.
Proof. Let e = (1, . . . , 1), the all ones vector, then q 1 (G) ≥ e T QGe e T e = 4m n with equality if G is a regular graph. Again, using (8) the inequality in the statement follows. If G is a regular bipartite graph then υ b (G) = 0 and q 1 (G) = 4m n thus, the equality in (10) follows.
Theorem 12 Let G be a connected graph with n vertices and vertex bipartite- Equality holds if G is a regular, bipartite graph.
The bounds in the previous theorems and corollaries show connections between signless Laplacian spread and the vertex bipartiteness v b (G). It is therefore natural to ask how difficult this parameter is to compute. The vertex bipartization problem is to find the minimum number of vertices in a graph whose deletion leaves a subgraph which is bipartite. This problem is NP-hard, even when restricted to graphs of maximum degree 3, see [10]. Actually, this problem has several applications, such as in via minimization in the design of integrated circuits ( [10]). Exact algorithms and complexity of different variants have been studied, see [33]. For the parameterized version, where k is fixed and one asks for k vertices whose deletion leaves a bipartite subgraph, an algorithm of complexity O(3 k · kmn) was found in [34]. Similarly, it is NP-hard to compute the edge bipartiteness ǫ b (G), even if all degrees are 3, see ( [10]). Finally, more general vertex and edge deleting problems were studied in [38], and NP-completeness of a large class of such problems was shown.
Recall that an induced subgraph is determined by its vertex set. In fact, deleting some vertices of G together with the edges incident on those vertices we obtain an induced subgraph. A set of vertices that induces an empty subgraph is called an independent set. The number of vertices in a maximum independent set of G is called the independence number of G and it is denoted by α (G). The problem of finding the independence number of a graph G is also NP-hard, see [14,24], whereas the spectral bounds can be determined in polynomial time.
Lemma 13 Let G be a graph with n vertices and independence number α (G).
Proof. Let S ⊆ V (G) be an independent set of vertices with cardinality α = α (G) and H be an induced subgraph of G such that V (H) = V (G) \S.
The adjacency matrix of G becomes where 0 is the square zero matrix of order α. Note that the cardinality of the set of edges of H satisfies The result is obtained since deleting all the edges of H yields a bipartite graph from G.
Proof. Let e = (1, 1, . . . , 1) be the all ones vector, then with equality if G is a regular graph. On the other hand, the condition in (11) implies that 4m Thus, by Lemma 13 By Theorem 12 the first inequality is obtained. The second inequality follows from the fact that Remark 15 Note that if α (G) = n−k and as m ≤ n(n−1) 2 a necessary condition for (11) is where τ (G) is the vertex cover number of G (that is the size of a minimum vertex cover in a graph G). Finding a minimum vertex cover of a general graph is an NP-hard problem however, for the bipartite graphs, the vertex cover number is equal to the matching number. Therefore, from the previous remark a necessary condition for (11) is, in this case, 4(n − 1) ≥ τ (G)(τ (G) − 1). Now, using Theorems 2 and 3 we derive the following results.
Theorem 16 Let G be a graph of order n with largest vertex degree ∆ and smallest vertex degree δ. If ∆ − δ ≥ 2, then . and otherwise (when ∆ − δ ≤ 1) Equality holds for G ≃ K 2 .
Proof. Let Q G = (q ij ) be the signless Laplacian matrix of G, then Q G is an n × n normal matrix and by Theorem 2 In this maximization we may assume (by symmetry) that d j ≥ d i . Moreover, by fixing d j − d i to some number k ∈ {0, 1, . . . , ∆ − δ}, we get is a convex quadratic polynomial in k so its maximum over k ∈ {0, 1, . . . , ∆ − δ} occurs in one of the two endpoints. Therefore Theorem 17 Let G be a graph of order n with largest vertex degree ∆ and smallest vertex degree δ.
We have equality for G ≃ K 2 and G ≃ K 1,3 .
Proof. Let Q(G) = (q ij ) be the signless Laplacian matrix of G, then Q G is an n × n symmetric matrix and by Theorem 3 we derive , and e ij and f ij are given in Theorem 3.
The next Corollary is a direct consequence of the previous theorem.
Let G be a graph with vertex degrees d 1 , d 2 , . . . , d n . Let be the first Zagreb index [18]. In [30] the following inequality related to the Cauchy-Schwarz inequality is shown. It follows directly from the Lagrange identity (see [36] concerning Lagrange identity and related inequalities).
Lemma 19 [30] Let a = (a 1 , a 2 , . . . , a n ) and b = (b 1 , b 2 , . . . , b n ) be two vectors with 0 < m 1 ≤ a i ≤ M 1 and 0 < m 2 ≤ b i ≤ M 2 , for i = 1, 2, . . . , n, for some constants m 1 , m 2 , M 1 and M 2 . Then By using the above result in what follows, we will obtain a lower bound for the s Q (G) in terms of M 1 (G), n and m.
Theorem 20 Let G be a connected graph with n ≥ 2 vertices. Then Proof. In this proof we use Lemma 19 with a i = 1 and b i = q i , for 1 ≤ i ≤ n. Since 0 < 1 ≤ a i ≤ 1, and 0 < q n ≤ b i ≤ q 1 , 1 ≤ i ≤ n. Thus M 1 M 2 = 1q 1 and m 1 m 2 = 1q n . By Lemma 19 This gives 8m Remark 21 Note that for a k-regular graph G the lower bound given by Theorem 20 is 2 √ k. So, for regular graphs, it is worse than the other one given by Corollary 18.

Lower bounds based on a minmax principle
In this section we introduce a principle for finding several lower bounds for the signless Laplacian spread of a graph.
Let B n denote the unit ball in R n . The next theorem gives a lower bound on the spread of a real symmetric matrix A. The result is actually known in a slightly different form (see below), but we give a new proof of this inequality, using ideas from minmax theory.

Theorem 22
Let A be a real symmetric matrix of order n. Then Proof. Lemma 1 in [22] says that s(A) = 2 min t∈R |A − tI| where the minimum is over all t ∈ R (this follows easily from the spectral theorem). Therefore The inequality above follows from standard minmax-arguments. In fact, for any function f = f (x, t) defined on sets X and T , we clearly have inf for all x ∈ X and t ∈ T . The desired inequality is then obtained by taking the infimum over t in the last inequality, and then, finally, the supremum over x. The final equality in (14) follows as this is a least-squares problem in one variable t, for given x ∈ B n , so geometrically t is chosen so that tx is the orthogonal projection of Ax onto the line spanned by x. The desired result now follows from (14).
Below we rewrite the bound in the previous theorem. First, however, note from the proof that the bound in (14) expresses the following: for any unit vector x, twice the distance from Ax to the line spanned by x is a lower bound on the spread. Thus, the bound has a simple geometrical interpretation. This may be useful, in specific situations, in order to find an x which gives a good lower bound.
Now, a straightforward computation shows that so Theorem 22 says that Therefore this result is actually the result presented in [27,Theorem 4]. In [27] the authors state that this result, in fact, goes back to Bloomfield and Watson in 1975, [4, (5.3)], and it was rediscovered by Styan [37,Theorem 1]. See also [20, section 5.4] and [21].

Remark 23
From the equality case of Cauchy-Schwarz Theorem, the bound in (16) is equal to zero when the vector τ and y are a linear combination of the vector e.
We may now obtain different lower bounds on the signless Laplacian spread s Q (G), for a graph G, by applying Theorem 22 to the signless Laplacian matrix Q G and choosing some specific unit vector x, or a nonzero vector y, and use (16).
For instance, consider the simple choice x = e i , the ith coordinate vector.
which gives a short proof of the second bound (when ∆ − δ ≤ 1) in Theorem 16.
Another application of this principle is obtained by using x as the normalized all ones vector, which gives the following lower bound.

Corollary 24
Let G be a graph of order n. Then Proof. We consider (15) with A = Q G and x = (1/ √ n)e where e denotes the all ones vector. Then x T A 2 x = (1/n)e T Q 2 G e. Let d = (d 1 , d 2 , . . . , d n ) be the vector whose components are the vertex degrees. So A G e = d and Thus (15) gives Next, we apply Theorem 22 using the degree vector d = (d 1 , d 2 , . . . , d n ). This gives the following result; it follows directly from (16).

Corollary 25 Let G be a graph. Then
where Next, since G is a graph without isolated vertices, we may use (16) with the n-tuple of the reciprocal of the vertex degrees of G.
We now establish some other lower bounds on s Q (G) based on other principles.
Theorem 27 Let G be a graph with vector degrees d = (d 1 , d 2 , . . . , d n ). Moreover, consider d (2) = d Note that if G is a bipartite graph then Υ = 0 = q n (G).
Proof. In [29, Theorem 6] the following lower bound for the spread s(B) of an Hermitian matrix B = (b ij ) was shown Replacing B by Q = Q G one has By the Proof of Corollary 29 we get .
Moreover, from the Proof of Theorem 6 in [29] one sees that corresponds to the smaller eigenvalue of the 2 × 2 submatrix of B, and we will see that the minimum (for the case of Q) corresponds to the smaller eigenvalue of some 2 × 2 submatrix of Q with the form d p 1 1 d q . Two cases must be considered.
1. The submatrix is d p 1 1 d q . By a straightforward computation, of the mentioned eigenvalue, we obtain and consider the function x 2 +α , one easily sees that f ′ (x) < 0, so f (x) is strictly decreasing, thus the minimum can not be obtained for small degrees. Recall that the maximum vertex degree is denoted by ∆. We conclude that

The submatrix is
It is clear that its smaller eigenvalue is thus Υ = δ is the minimum vertex degree of G. We recall the above As the constant α in function f equals the negative number α = 1 − d p d q , we have Moreover, as the result follows.
Theorem 29 Let G be a k regular graph. Then Proof. Let G be a k-regular graph then, By the inequality in (18), one obtains = |2k − (k − 1)| = k + 1.
Thus the statement follows.
Remark 30 For k > 3 the previous lower bound improves the lower bound given in Corollary 18.

Upper bounds
In [6], using the Mirsky's upper bound mentioned above, it was shown that for a graph G with n ≥ 5 vertices and m ≥ 1 edges, the following inequality holds Here equality holds if and only if G is one of the graphs K n , G( n 4 , n 4 ), The graph G(r, s) is the graph obtained by joining each vertex of the subgraph K s of K r ∨K s to all the vertices of K s of another copy of K r ∨K s . Here G∨G ′ is the usual join operation between two graphs G and G ′ .
Theorem 31 Let G be an (n, m)-graph. Then The equality is attained if and only if G ≃ K n 2 , n 2 .
Proof. Since Q = Q(G) is a normal matrix, by applying Theorem 1 to Q we obtain with equality if and only if the eigenvalues q 1 , q 2 , . . . , q n satisfying the following condition ( * ) q 2 = q 3 = · · · = q n−1 = q 1 + q n 2 .
As Q 2 F = M 1 (G) + 2m and tr Q = 2m, the result follows. If condition ( * ) holds then tr Q = nq 2 so by [35,Lemma 1.1]. Then q 1 + q n = 2q 2 ≤ q 1 so q n = 0 and q 2 = 1 2 q 1 . This gives Thus, G is a regular bipartite graph and the statement holds. Conversely, if G ≃ K n 2 , n 2 , by a standard verification, the inequality in Theorem 31 holds with equality.
Corollary 32 Let G k be a k-regular graph with n vertices. Then Here equality is attained if and only if G ≃ K n 2 , n 2 .
Proof. We get M 1 (G k ) = nk 2 and m = nk 2 . Thus, By Theorem 31, the result now follows. If G ≃ K n 2 , n 2 , then G is a regular bipartite graph with k = n 2 , so √ 2nk = 2n n 2 = n = µ 1 (K n 2 , n 2 ) = s Q (K n 2 , n 2 ). Corollary 33 Let G be an (n, m)-graph. Then The equality is attained if and only if G ≃ K n 2 , n 2 . Proof. In [7] it was shown that with equality if and only if G is either a star, a regular graph or a complete graph K ∆+1 with n − ∆ − 1 isolated vertices. Replacing M 1 (G) in (20) by its upper bound in (22) the result follows. Equality holds in (21) if and only if equality holds in both (20) and (22), or equivalently G ≃ K n

Comparison of bounds
This section deals with a comparison of some of the bounds presented in this work. We firstly compare the bound in Theorem 17 with the lower bound for s Q (G) (depending on same parameters) found in [26, Corollary 2.3]: Let L 1 (G) and L 2 (G) denote the bound from Theorem 17 and [26], respectively, so Observe that L 1 (G) only depends on the minimum and maximum degrees, not n and m. Letd = (1/n) n i=1 d i = 2m/n denote the average degree in G. So which shows that L 2 (G) is determined by the maximum and average degree as well as n. Hered − ∆ ≤ 0, andd − ∆ = 0 precisely when G is regular. The next result relates the two lower bounds as a function of certain graph properties.
Theorem 34 Let G be a an (n, m)-graph with n > 2.
(ii) Assume G is connected and contains a pendant vertex. Then L 2 (G) ≤ n n−1 ∆ and L 1 (G) = √ ∆ 2 + 7. In particular, L 2 (G) < L 1 (G) holds if Proof. (i) The two expressions follow from the calculation above asd = δ = ∆ = k. Consider the case when G is regular, say of degree k. Then Here the right hand side is greater than 1 precisely when k ≥ 5, and the conclusion then follows.
Note that the lower bound L 1 (G) for the regular case is worse than the lower bound in Theorem 29.
Consider again our lower bound on the signless Laplacian spread s Q (G) In the proof we obtained the bound by some calculations in which a single inequality was involved, namely when we used that "minmax" is at least as large as "maxmin", for the function involved. Unfortunately, we cannot show that equality holds here. The reason for this is basically that (Q − tI n )x is not a concave function of x and, also, B n is not a convex set, so general minmax theorems may not be applied to our situation. However, it is interesting to explore further the quality of the best bound one gets from the minmax principle. To do so, consider the function . Note that f is a complicated function, obtained from a multivariate polynomial of degree six (by taking the square root, although that can be removed for the maximization). We consider an extremely simple approach to approximately maximize f over the unit ball; we perform a few iterations K of the following gradient method with a step length s > 0: Algorithm: Simple gradient search.
In each iteration, we make a step in the direction of the (numerical) gradient, even if the new function value could be less. Thus we avoid line search. The disadvantage is that we may not approximate a local maximum so well, but the advantage is that we can escape a local maximum and go towards another with higher function value. The procedure is very simple, and heuristic, and we typically only perform a few iterations K (around 10 or 20). We have used constant step length s, but also variable step length (being a decreasing function of the iteration number).
In the table below we give some computational results, for 5 random, connected graphs, showing all previous lower bounds we have discussed and the new bound η. The notation in the table is the following:  For the last example above we next show the value of η during the 10 iteration of the gradient search algorithm, and we see that that maximum, in this case, was found in iteration 4: These, and similar, experiments clearly show that η(G) is a very good lower bound on the signless Laplacian spread s Q (G). Although the exact computation of η(G) may be hard, we see that a simple gradient algorithm finds very good approximations, and lower bounds on s Q (G), in a few iterations. Of, course, the result of such an algorithm is not an analytical bound in terms of natural graph parameters. But every bound needs to be computed, and, in practice, its computational effort should always be compared to the work of using an eigenvalue algorithm for computing the largest and smallest eigenvalue of Q, and finding s Q (G) in that way.
Finally, we remark that it is possible to use the results above to find such an analytical bound which is quite good: compute the exact gradient (of f (x) 2 ) at the constant vector and make one iteration in the gradient algorithm; let x be the obtained unit vector, and compute the bound f (x). We leave this computation to the interested reader.