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Let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0 be the normalized Laplacian eigenvalues of a graph G with n vertices. Also, let χ and α be the chromatic number and the independence number of a graph G, respectively. In this paper, we discuss some properties of graphs with ρ 1 = χ / (χ − 1). In particular, we characterize all the graphs with ρ 1 = χ / (χ − 1) when the maximum degree is n−1. Moreover, we obtain an upper bound on the multiplicity of normalized Laplacian eigenvalues m G (ρ) in terms of n and α, and also characterize graphs for which the bound is attained. [ABSTRACT FROM AUTHOR]

Let , and be, respectively, the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix of graph G, where denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of and will be denoted by and , respectively. In this paper, we study the properties of the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues of graph G. [ABSTRACT FROM AUTHOR]

Abstract Let G = (V , E) be a simple graph of order n with normalized Laplacian eigenvalues ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0. The normalized Laplacian spread of graph G , denoted by ρ 1 − ρ n − 1 , is the difference between the largest and the second smallest normalized Laplacian eigenvalues of graph G. In this paper, we obtain the first four smallest values on ρ 2 of graphs. Moreover, we give a lower bound on ρ 2 of connected bipartite graph G except the complete bipartite graph and characterize graphs for which the bound is attained. Finally, we present some bounds on the normalized Laplacian spread of graphs and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n . The energy E ( G ) of a simple graph G is the sum of absolute values of the eigenvalues of its adjacency matrix. The Laplacian energy, the signless Laplacian energy and the distance energy of graph G are denoted by L E ( G ) , S L E ( G ) and D E ( G ) , respectively. In this paper we introduce a distance Laplacian energy D L E and distance signless Laplacian energy D S L E of a connected graph. We present Nordhaus–Gaddum type bounds on Laplacian energy L E ( G ) and signless Laplacian energy S L E ( G ) in terms of order n of graph G and characterize graphs for which these bounds are best possible. The complete graph and the star give the smallest distance signless Laplacian energy D S L E among all the graphs and trees of order n , respectively. We give lower bounds on distance Laplacian energy D L E in terms of n for graphs and trees, and characterize the extremal graphs. Also we obtain some relations between D E , D S L E and D L E of graph G . Moreover, we give several open problems in this paper. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n with m edges. Denote by D ( G ) the diagonal matrix of its vertex degrees and by A ( G ) its adjacency matrix. Then the Laplacian matrix of graph G is L ( G ) = D ( G ) − A ( G ) . Among all eigenvalues of the Laplacian matrix L ( G ) of graph G , the most studied is the second smallest, called the algebraic connectivity a ( G ) of a graph. Let d ‾ ( G ) and δ ( G ) be the average degree and the minimum degree of graph G , respectively. In this paper we characterize all graphs for which ( i ) a ( G ) = 1 with δ ( G ) ≥ ⌈ n − 1 2 ⌉ , and ( i i ) a ( G ) = 2 with δ ( G ) ≥ n 2 . In [1] , Aouchiche mentioned a conjecture involving the algebraic connectivity a ( G ) and the average degree d ‾ ( G ) of graph G : a ( G ) − d ‾ ( G ) ≥ 4 − n − 4 n with equality holding if and only if G ‾ ≅ K 1 , n − 2 ∪ K 1 ( K 1 , n − 2 is a star of order n − 1 and G ‾ is the complement of graph G ). Here we prove this conjecture. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix and the signless Laplacian matrix of G are L(G)=D(G)-A(G) and Q(G)=D(G)+A(G), respectively. Also denote by λ1(G), a(G), q1(G) and δ(G) the largest eigenvalue of A(G), the second smallest eigenvalue of L(G), the largest eigenvalue of Q(G) and the minimum degree of G, respectively. In this paper, we give partial proofs to the following two conjectures: (i)Aouchiche (Comparaison Automatisée d’Invariants en Théorie des Graphes, 2006) if G is a connected graph, then a(G)/δ(G) is minimum for graph composed of 2 triangles linked with a path. (ii)Aouchiche et al. (Linear Algebra Appl 432:2293-2322, 2010) and Cvetković et al. (Publ Inst Math Beogr 81(95):11-27, 2007) if G is a connected graph with n≥4 vertices, then q1(G)-2λ1(G)≤n-2n-1 with equality holding if and only if G≅K1,n-1. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex set V and edge set E. Let di be the degree of the vertex vi of G. The inverse degree, Randić index, and harmonic index of G are defined as ID =Σvi∈V 1/di, R =Σvivj∈E 1/√di dj, and H = Σvivj∈E 2/(di + dj), respectively. We obtain relations between ID and R as well as between ID and H. Moreover, we prove that in the case of trees, ID > R and ID > H. [ABSTRACT FROM AUTHOR]

The aim of this note is to establish new spectral bounds for the harmonic matrix. [ABSTRACT FROM AUTHOR]

We study the Laplacian and the signless Laplacian energy of connected unicyclic graphs, obtaining a tight upper bound for this class of graphs. We also find the connected unicyclic graph on n vertices having largest (signless) Laplacian energy for 3 ≤ n ≤ 13 . For n ≥ 11 , we conjecture that the graph consisting of a triangle together with n − 3 balanced distributed pendent vertices is the candidate having the maximum (signless) Laplacian energy among connected unicyclic graphs on n vertices. We prove this conjecture for many classes of graphs, depending on σ , the number of (signless) Laplacian eigenvalues bigger than or equal to 2. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n with diameter d . Remoteness ρ of G is the maximum average distance from a vertex to all others and ∂ 1 ≥ ⋯ ≥ ∂ n are the distance eigenvalues of G . Aouchiche and Hansen (0000), Aouchiche and Hansen conjectured that ρ + ∂ 3 > 0 when d ≥ 3 and ρ + ∂ ⌊ 7 d 8 ⌋ > 0 . In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on ∂ n + ρ and ∂ 1 − ρ when G ≇ K n and the extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n with m edges and clique number ω . Let μ 1 ≥ μ 2 ≥ … ≥ μ n = 0 be the Laplacian eigenvalues of G and let σ = σ ( G ) ( 1 ≤ σ ≤ n ) be the largest positive integer such that μ σ ≥ 2 m n . In this paper we study the relation between σ and ω . In particular, we provide the answer to Problem 2.3 raised in Pirzada and Ganie (2015) [15] . Moreover, we characterize all connected threshold graphs with σ < ω − 1 , σ = ω − 1 and σ > ω − 1 . We obtain Nordhaus–Gaddum-type results for σ . Some relations between σ with other graph invariants are obtained. [ABSTRACT FROM AUTHOR]

The k-th power of a graph G, denoted by [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a connected graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The transmission Tr ( v i ) of vertex v i is defined to be the sum of distances from v i to all other vertices. Let Tr ( G ) be the n × n diagonal matrix with its ( i , i ) -entry equal to Tr G ( v i ) . The distance signless Laplacian is defined as D Q ( G ) = Tr ( G ) + D ( G ) , where D ( G ) is the distance matrix of G . Let ∂ 1 ( G ) ≥ ∂ 2 ( G ) ≥ ⋯ ≥ ∂ n ( G ) denote the eigenvalues of distance signless Laplacian matrix of G . In this paper, we first characterize all graphs with ∂ n ( G ) = n − 2 . Secondly, we characterize all graphs with ∂ 2 ( G ) ∈ [ n − 2 , n ] when n ≥ 11 . Furthermore, we give the lower bound on ∂ 2 ( G ) with independence number α and the extremal graph is also characterized. [ABSTRACT FROM AUTHOR]

The energy E ( G ) of a simple graph G is the sum of absolute values of the eigenvalues of its adjacency matrix. The line graph of G is denoted by L G . We obtain a relation between E ( L G ) and E ( G ) , and establish bounds on E ( L G ) . Moreover, we present results pertaining to equienergetic and hyperenergetic graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected threshold graph of order n with m edges and trace T . In this paper we give a lower bound on Laplacian energy in terms of n, m and T of G . From this we determine the threshold graphs with the first four minimal Laplacian energies. Moreover, we obtain the threshold graphs with the largest and the second largest Laplacian energies. [ABSTRACT FROM AUTHOR]

Let G be a bipartite graph of order n with m edges. The energy E ( G ) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A . In 1974, one of the present authors established lower and upper bounds for E ( G ) in terms of n, m , and det A . Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m , and the first Zagreb index is obtained, and the extremal graphs characterized. [ABSTRACT FROM AUTHOR]

The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G , defined as D Q ( G ) = T r ( G ) + D ( G ) , where D ( G ) is the distance matrix of G and T r ( G ) is the diagonal matrix of vertex transmissions of G . In this paper we determine upper and lower bounds on the minimal and maximal entries of the principal eigenvector of D Q ( G ) and characterize the extremal graphs. In addition, we obtain a lower bound on the distance signless Laplacian spectral radius of G based on its order and independence number, and characterize the extremal graph. [ABSTRACT FROM AUTHOR]

For G being a graph with n vertices and m edges, and with Laplacian eigenvalues μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n − 1 ≥ μ n = 0 , the Laplacian energy is defined as L E = ∑ i = 1 n | μ i − 2 m / n | . Let σ be the largest positive integer such that μ σ ≥ 2 m / n . We characterize the graphs satisfying σ = n − 1 . Using this, we obtain lower bounds for LE in terms of n, m , and the first Zagreb index. In addition, we present some upper bounds for LE in terms of graph invariants such as n, m , maximum degree, vertex cover number, and spanning tree packing number. [ABSTRACT FROM AUTHOR]

For any real number α , let s α ( G ) denote the sum of the α th power of the non-zero Laplacian eigenvalues of a graph G . In this paper, we first obtain sharp bounds on the largest and the second smallest Laplacian eigenvalues of a graph, and a new spectral characterization of a graph from its Laplacian eigenvalues. Using these results, we then establish sharp bounds for s α ( G ) in terms of the number of vertices, number of edges, maximum vertex degree and minimum vertex degree of the graph G , from which a Nordhaus–Gaddum type result for s α is also deduced. Moreover, we characterize the graphs maximizing s α for α > 1 among all the connected graphs with given matching number. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G , defined as Q ( G ) = Tr ( G ) + D ( G ) , where Tr ( G ) is the diagonal matrix of vertex transmissions of G and D ( G ) is the distance matrix of G . In [10] , Xing et al. determined the graph with minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. For n ≥ 3 , let T n − 3 , 1 1 be the n -vertex tree of maximum degree n − 2 . In this paper, we show that T n − 3 , 1 1 gives the second minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. Moreover, we prove two conjectures involving the second largest eigenvalue of the distance signless Laplacian matrix Q ( G ) of graph G . [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with vertex set V = { v 1 , v 2 , … , v n } . Also let μ 1 ( G ) ≥ μ 2 ( G ) ≥ ⋯ ≥ μ n − 1 ( G ) ≥ μ n ( G ) = 0 and q 1 ( G ) ≥ q 2 ( G ) ≥ ⋯ ≥ q n ( G ) ≥ 0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G , respectively. In this paper we obtain μ i ( G ) ≤ i − 1 + min U i ⁡ max ⁡ { | N H ( v k ) ∪ N H ( v j ) | : v k v j ∈ E ( H ) } , where N H ( v k ) is the set of neighbors of vertex v k in V ( H ) = V ( G ) \ U i , U i is any ( i − 1 ) -subset of V ( G ) (here, we agree that i ∈ { 1 , … , n − 1 } and μ i ( G ) ≤ i − 1 if E ( H ) = ∅ ). For any graph G , this bound does not exceed the order of G . Moreover, we prove that max ⁡ { μ i ( G ) , q i ( G ) } ≤ max i ≤ k ≤ n ⁡ { d G ( v k ) + ∑ v j ∈ N G ( v k ) ∩ N d G ( v j ) d G ( v k ) } ≤ 2 d G ( v i ) , where d G ( v i ) is the i -th largest degree of G and N = { v i , v i + 1 , … , v n } . [ABSTRACT FROM AUTHOR]

Abstract: Let be a graph with vertices and edges. Also let be the eigenvalues of the Laplacian matrix of graph . The Laplacian energy of the graph is defined as In this paper, we present some lower and upper bounds for of graph in terms of , the number of edges and the maximum degree . Also we give a Nordhaus–Gaddum-type result for Laplacian energy of graphs. Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs. [Copyright &y& Elsevier]

Abstract: Let G be a connected graph of order n with Laplacian eigenvalues . The Laplacian-energy-like invariant of the graph G is defined as Lower and upper bounds for LEL are obtained, in terms of n, number of edges, maximum vertex degree, and number of spanning trees. [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The Laplacian matrix of G is , where is the diagonal matrix of its vertex degrees and is the adjacency matrix. Let be the Laplacian eigenvalues of G. For a graph G and a real number , the graph invariant is the sum of the β-th power of the non-zero Laplacian eigenvalues of G, that is, In this paper, we obtain some lower and upper bounds on for G in terms of n, the number of edges m, maximum degree , clique number ω, independence number α and the number of spanning trees t. Moreover, we present some Nordhaus–Gaddum-type results for of G. [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The Laplacian matrix of G is , where is the diagonal matrix of its vertex degrees and is the adjacency matrix. Among all eigenvalues of the Laplacian matrix of a graph, the most studied is the second smallest, called the algebraic connectivity of a graph [18]. In this paper we give a lower bound on the algebraic connectivity of graphs. Moreover, we mention two conjectures, obtained by the system AutoGraphiX, about the algebraic connectivity , diameter and the minimum degree of graphs (see, [2], available online at http://www.gerad.ca/∼agx/):with equality if and only if G is isomorphic to a graph with and , and (ii) is minimum for a graph composed of 2 cliques on vertices with a common vertex if n is odd, and linked by an edge if n is even. Here we prove conjecture in (i) using the lower bound on the algebraic connectivity of graphs and conjecture in (ii), respectively. [Copyright &y& Elsevier]

Abstract: The power of a graph , denoted by , is the graph with the same vertex set as , such that two vertices are adjacent in if and only if their distance is at most in . In this paper, we give bounds on the first two largest Laplacian eigenvalues of the second power of a general graph, and on the second power of a tree. We also give a Nordhaus–Gaddum-type inequality for the Laplacian spectral radius of . [Copyright &y& Elsevier]

Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]

Abstract: Let G be a connected graph of order n with Laplacian eigenvalues . The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as and , respectively. We compare Kf and LEL and establish two sufficient conditions under which . The connected graphs of order n with nine greatest Kirchhoff indices are determined; for these holds. [Copyright &y& Elsevier]

Abstract: Let be a simple graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of is . In , Cvetković et al. (2007) have given conjectures on signless Laplacian eigenvalues of (see also Aouchiche and Hansen (2010) , Oliveira et al. (2010) ). Here we prove two conjectures. [Copyright &y& Elsevier]

Abstract: Let G =(V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G)= D(G)+ A(G). In [5], Cvetković et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue () and the index () of graph G (see also Aouchiche and Hansen [1]):with equality if and only if G is the star for the lower bound, and if and only if G is the complete bipartite graph for the upper bound. In this paper we prove the lower bound and characterize the extremal graphs. [Copyright &y& Elsevier]

Abstract: We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained. [Copyright &y& Elsevier]

Abstract: Let be a simple graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the Laplacian matrix of is and the signless Laplacian matrix of is . In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of . In , Cvetković et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of (see also ). Here we prove five conjectures. [Copyright &y& Elsevier]

Abstract: Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on are given. We show that some known bounds can be obtained from our bounds. [Copyright &y& Elsevier]

Abstract: Let be a simple graph of order with edges. Let the adjacency spectrum be of , where . The Estrada index of a graph is . In [J.A. Peña, I. Gutman, J. Rada, Estimating the Estrada index, Linear Algebra Appl. 427 (2007) 70–76], Peña et al. posed a conjecture that the star has maximum Estrada index for any tree of order and the path has minimum Estrada index for any tree of order or any connected graph of order . In this paper, we have proved that the star has maximum Estrada index for any tree. Also, we obtain that the path has minimum Estrada index for any connected graph with or . Moreover, we give better lower bound on Estrada index for any connected graph. [Copyright &y& Elsevier]

Abstract: We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained. [Copyright &y& Elsevier]

Let G = ( V, E) be a simple graph with n vertices, e edges and d 1 be the highest degree. Further let λ i , i = 1,2,..., n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ2 = λ3 = ... = λ n-1 if and only if G is a complete graph or a star graph or a ( d 1, d 1) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d 1 only: The equality holds if and only if G is a star graph or a complete graph. Earlier bounds by Grimmett [5], Grone and Merris [6], Nosal [11], and Kelmans [2] were sharp for complete graphs only. Also our bound depends on n, e and d 1 only. [ABSTRACT FROM AUTHOR]

In this article, we present lower bounds for the largest eigen value, the second largest eigenvalue and the sum of the two largest eigenvalues of the Laplacian matrix of a graph. [ABSTRACT FROM AUTHOR]

Let λ1(T) and λ2(T) be the largest and the second largest Laplacian eigenvalues of a tree T. We obtain the following sharp lower bound for λ1(T):λ1(T)&ges;max(di+mi+1)+(di+mi+1)2−4(dimi+1)/2:vi∈V,wheredi and mi denote the degree of vertex vi and the average of the degrees of the vertices adjacent to vertex vi respectively. Equality holds if and only if T is a tree T(di,dj), where T(di,dj) is formed by joining the centres of di copies of K1,dj−1 to a new vertex vi, that is, T(di,dj)−vi=diK1,dj−1.

Let v1 be the highest degree vertex of degree d1 and v2 be the second highest degree vertex of degree d2. We also show that if T is a tree of order n>2, thenλ2(T)&ges;d2ifv1v2∈E,(d2+1)+(d2+1)2−4/2: vi∈V,wheredi and mi denote the degree of vertex vi and the average of the degrees of the vertices adjacent to vertex vi respectively. Equality holds if and only if T is a tree T(di,dj), where T(di,dj) is formed by joining the centres of di copies of K1,dj−1 to a new vertex vi, that is, T(di,dj)−vi=diK1,dj−1.Let v1 be the highest degree vertex of degree d1 and v2 be the second highest degree vertex of degree d2. We also show that if T is a tree of order n>2, thenλ2(T)&ges;d2if v1v2∈E,(d2+1)+(d2+1)2−4/2if v1v2∉E,whereE is the set of edges. Equality holds if T=T1(d1) or T=T2(d1), where T1(d1) is formed by joining the centres of two copies of K1,d1−1 and T2(d1) is formed by joining the centres of two copies of K1,d1−1 to a new vertex.Moreover, we obtain the lower bounds for the sum of two largest Laplacian eigenvalues. [Copyright &y& Elsevier]

Let G=(V,E) be a simple connected graph and λ1(G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: [Copyright &y& Elsevier]

Let a ( k ) = ( a 1 , a 2 , … , a k ) be a sequence of positive integers. A butterfly-like graph W p ( s ) ; a ( k ) is a graph consisting of s ( ≥ 1 ) cycle of lengths p + 1 , and k ( ≥ 1 ) paths P a 1 + 1 , P a 2 + 1 , …, P a k + 1 intersecting in a single vertex. The girth of a graph G is the length of a shortest cycle in G . Two graphs are said to be A - cospectral if they have the same adjacency spectrum. For a graph G , if there does not exist another non-isomorphic graph H such that G and H share the same Laplacian (respectively, signless Laplacian) spectrum, then we say that G is L − D S (respectively, Q − D S ). In this paper, we firstly prove that no two non-isomorphic butterfly-like graphs with the same girth are A -cospectral, and then present a new upper and lower bounds for the i -th largest eigenvalue of L ( G ) and Q ( G ) , respectively. By applying these new results, we give a positive answer to an open problem in Wen et al. (2015) [17] by proving that all the butterfly-like graphs W 2 ( s ) ; a ( k ) are both Q − D S and L − D S . [ABSTRACT FROM AUTHOR]