On the normalized Laplacian energy and general Randić index R 1 of graphs

Transcription

1 On the normalized Laplacian energy and general Randić index R of graphs Michael Cavers a Shaun Fallat a Steve Kirkland ab3 a Department of Mathematics and Statistics University of Regina Regina SK Canada S4S 0A b Hamilton Institute National University of Ireland Maynooth Co Kildare Ireland Abstract In this paper we consider the energy of a simple graph with respect to its normalized Laplacian eigenvalues which we call the L-energy Over graphs of order n that contain no isolated vertices we characterize the graphs with minimal L-energy of and maximal L-energy of n/ We provide upper and lower bounds for L-energy based on its general Randić index R (G) We highlight known results for R (G) most of which assume G is a tree We extend an upper bound of R (G) known for trees to connected graphs We provide bounds on the L-energy in terms of other parameters one of which is the energy with respect to the adjacency matrix Finally we discuss the maximum change of L-energy and R (G) upon edge deletion Key words: normalized Laplacian matrix graph energy general Randić index AMS Classification: 05C50; 05C90; 5A8; 5A4 Introduction Throughout this paper all graphs are simple and have no isolated vertices We will use d G x to denote the degree of a vertex x in G If there is only one graph in question we simply write d x Let A be the adjacency matrix of a graph G and D be the diagonal matrix of vertex degrees The normalized Laplacian matrix of a graph G denoted by L is defined to be the matrix with entries if x = y and d y 0 L(x y) = dx d y if x and y are adjacent in G 0 otherwise Note that L has the following relationship to A and D: L = I D / AD / Corresponding author addresses: caversm@mathureginaca (Michael Cavers) sfallat@mathureginaca (Shaun Fallat) stephenkirkland@nuimie (Steve Kirkland) Research supported in part by an NSERC PGS D Research supported in part by an NSERC Research Grant 3 Research supported in part by an NSERC Discovery Grant Preprint submitted to Elsevier April 8 00

2 It is well known that 0 is an eigenvalue of L and that the remaining eigenvalues lie in the interval [0 ] (see [3] for other properties of the eigenvalues of L) For convenience if M is a real symmetric matrix of order n we order and denote the eigenvalues by λ (M) λ n (M) and the singular values by σ (M) σ n (M) If G is a graph and M is a real symmetric matrix associated with G then the M-energy of G is n E M (G) = λ i(m) tr(m) n () i= where tr(m) is the trace of M Gutman [8] introduced the energy of a graph in 978 Recently the adjacency energy [9] Laplacian energy [] signless Laplacian energy distance energy [] and incidence energy [0] of a graph has received much interest Along the same lines the energy of more general matrices and sequences has been studied (see [ 7]) The goal of this paper is to analyze the L-energy of a graph and determine how graph structure relates to L-energy Formally using () with M taken to be L the normalized Laplacian energy (or L-energy) of a graph G is E L (G) = It is easy to see that this is equivalent to E L (G) = = n λ i (L) i= n λ i (I L) () i= n σ i (I L) (3) It should be noted that Nikiforov [7] defines the energy of a matrix M of order n to be E(M) = i= n σ i (M) i= in which case by (3) we are interested in E(I L) In this paper we use the M-energy definition in () when referring to the energy of a real symmetric matrix Let G be a graph of order n (with no isolated vertices) A convenient parameter of G is the general Randić index R α (G) defined as R α (G) = x y (d x d y ) α (4) where the summation is over all (unordered) edges xy in G and α 0 is a fixed real number In 975 Randić [] proposed a topological index R (with α = ) under the name branching index In 998 Bollobás and Erdős [] generalized this index by replacing the / with any real number α (as defined in (4)) The papers [4 5] survey recent results on the general Randić index of graphs with an emphasis on trees and chemical graphs In Section we will see that the L-energy of G can be bounded in terms of R (G) We then highlight some relevant results on the parameter R (G) that appear in the literature

3 We provide an upper bound on R (G) in the case that G is a connected graph Finally we discuss how R (G) changes when an edge is deleted In Section 3 we show n E L (G) and characterize the graphs attaining these bounds If G is connected then the upper bound on the L-energy can be improved to E L (G) < (n + ) We provide a class of connected graphs attaining L-energy E L (G) = n + O() and ask if this class has maximal L-energy over all connected graphs Finally we discuss other bounds for E L (G) and how edge deletion affects L-energy 5 8 Bounds on R (G) and its relationship to L-energy We begin with two other formulations of R (G) By analyzing the entries in (I L) and using (4) with α = observe that By rewriting (4) with α = we get R (G) = tr((i L) ) (5) R (G) = y V (6) d y d x x x y where f(x y) x x y represents the sum over all (unordered) edges xy in G that are incident to a fixed vertex y in the vertex set V of G Using (4) the quantity R (G) can be found by putting a weight of d xd y on each edge xy of G (which we call the weight of edge xy) and then summing the weights over all the edges of G Alternatively using (6) R (G) can be found by putting a weight of d y x x y on each vertex y of G (which we call the weight of vertex y) and then summing the weights over all the vertices of G In the next lemma we see the importance of R (G) when analyzing the L-energy of a graph Lemma Let G be a graph of order n with no isolated vertices Then d x R (G) E L (G) nr (G) 3

4 Proof By the Cauchy-Schwartz inequality with () (using vectors ( ) T and ( λ (I L) λ n (I L) ) T ) along with (5) we obtain the upper bound E L (G) n n [λ i (I L)] = n tr((i L) ) = nr (G) i= Note that the eigenvalues of I L lie in the interval [ ] Thus [λ i (I L)] λ i (I L) giving E L (G) = n λ i (I L) i= n [λ i (I L)] = tr((i L) ) = R (G) i= Thus determining how the structure of a graph relates to R (G) will provide information about E L (G) In the remainder of this section we look at bounds on R (G) We first highlight a few known results that can be found in the literature By considering the minimum and maximum degrees of G Shi [3] has obtained upper and lower bounds for R (G) Theorem [3 Theorem & 3] Let G be a graph of order n with no isolated vertices Suppose G has minimum vertex degree equal to d min and maximum vertex degree equal to d max Then n R (G) n d max d min Equality occurs in both bounds if and only if G is a regular graph Li and Yang [6] provide bounds on R (G) given strictly in terms of the order of G Note that the length of a path is the number of edges that the path uses Theorem 3 [6 Theorem 3] Let G be a graph of order n with no isolated vertices Then n n (n ) R (G) with equality in the lower bound if and only if G is a complete graph and equality in the upper bound if and only if either (i) n is even and G is the disjoint union of n/ paths of length or (ii) n is odd and G is the disjoint union of (n 3)/ paths of length and one path of length If G is a disconnected graph with k connected components in particular G G G k then k R (G) = R (G i ) i= Thus it is interesting to know how R (G) behaves for the class of connected graphs In [4] Clark and Moon provide bounds on R (T ) for a tree T of order n They showed that R (T ) 5n

5 In [3] Hu Li and Yuan refine this upper bound however gaps were found in their proof (see [8]) Then Pavlović Stojanvoić and Li gave a sound proof in [9] Theorem 4 [3 9] For a tree T of order n 03 R (T ) 5n See [0] for a further refinement (which we omit here) giving a sharp upper bound for R (T ) amongst all trees T of order n for n 70 Also see [4 5] for many other results concerning bounds for R (T ) In what follows we will see that the bound R (G) 5(n+) holds for any connected graph G of order n 3 We say G has a suspended path from u to w if uvw is a path with d G u = and d G v = Note that we don t require d G w 3 as in [4] A (t s + t)-system centered at r is an induced subgraph of G such that there are t suspended paths to vertex r and d G r = s + t This is illustrated in Figure s x x r x t y y y t t Figure : A (t s + t)-system centered at r A (k t s+k)-system centered at R is an induced subgraph of G that has k vertex disjoint (t t+)-systems centered at r r r k such that R is adjacent to each r i and d G R = s+k This is illustrated in Figure s R x x r x t r k xk x k x k t y y y t y k y k y t k t t k Figure : A (k t s + k)-system centered at R 5

6 The set of all (t s + t)-systems of G for s 0 and t and (k t s + k)-systems of G for s 0 and k t is referred to as the collection of systems of G Any object in this collection is referred to as a system of G Note that a vertex z of G may be the center of many different systems One question to ask is if there is always a tree on n vertices that maximizes R (G) over all connected graphs of order n If the answer is yes then the bound for connected graphs would follow immediately A first approach would be to look at the spanning trees of G and see if R (G) R (T ) for some spanning tree T of G However it is interesting to note that there exist graphs G such that for every spanning tree T of G the inequality R (T ) < R (G) holds Let G be the graph described as follows: Let t > 6 be a natural number and consider a cycle with 3 vertices a a a 3 each with degree 4 and with each of a a a 3 being the center of a ( t 4)-system The order of G is t + 9 and the only spanning trees of G are obtained by removing an edge on the cycle (namely a a a a 3 or a a 3 ) If T is any spanning tree of G then R (G) R (T ) = ( t + 3 ) ( 4 6 3t + ) 6 = t 6 48t Thus for t > 6 we have that for every spanning tree T of G R (T ) < R (G) To prove that R (G) 5(n+) for connected graphs G of order n 3 we take the same approach as done in the tree case An inductive argument will be used Let S be a subset of vertices of G We denote the graph obtained by deleting all the vertices in S and their incident edges by G\S We begin with an inequality relating R (G) to R (G\S) Note that deleting vertices (and edges) of G changes the degree sequence and so the weighted graph associated with G\S will not be an induced weighted subgraph of the weighted graph associated with G Observation 5 Let S be a subset of vertices of G then R (G) R (G\S) + x y x Sy S d G x d G y + x y xy S d G x d G y In [3] to prove the upper bound on R (T ) the edge weights of T were summed up at the end of the proof In general for connected graphs it is more beneficial to use the formulation (6) of R (G) and sum up the vertex weights (as seen in the final case of the proof below) Some of the cases in [3 9 0] can be extended to general graphs but for completeness of this paper we provide the full proof in the general case Note that in Cases (0)-(iii) we use /4 instead of 5/ in our manipulations of the second term Theorem 6 Let G be a connected graph on n 3 vertices Then R (G) 5(n + ) Proof The proof is by induction on the number of vertices If n = 3 then the path of length and the triangle both satisfy the inequality Let G be a connected graph on n 4 vertices and assume that the inequality holds for connected graphs on fewer than n vertices Case (0): If G has minimum degree at least then by Theorem we have R (G) n/4 and so the inequality holds 6

7 Case (i): Let x be a vertex of degree that is adjacent to a vertex y with d y 4 Deleting the vertex x does not disconnect the graph thus using S = {x} in Observation 5 along with induction we have R (G) R (G\{x}) + d y 5 n + 4 < 5 (n + ) Case (ii): Let z be a vertex of degree such that z x z y d x d y (a) Suppose x y in G and either d x = d y or d y d x Then form a graph H by deleting z and adding the edge xy Note that d H x = d x and d H y = d y Thus R (G) = R (H) + d x + d y d x d y Since H has n vertices and is connected we have by induction that R (G) 5 n + d x + d y d x d y < 5 (n + ) + d x + d y d x d y 4 4d x d y = 5 (n + ) + (d x )( d y ) 4d x d y If d x = d y or d y d x then R (G) < 5 (n + ) (b) Suppose x y in G (and hence d y d x ) then R (G) R (G\{z}) d x d y d x d y (d x )(d y ) Since deleting z does not disconnect the graph we have by induction that where R (G) < 5 (n + ) f(d x d y ) 4d x d y (d x )(d y ) f(x y) = (x )(x )y (3x + )(x )y + (x + )(x ) Our goal is to show that f(x y) 0 for y x (with x y integral) Note that f( y) = 8 for all y Fix x = x 0 3 and view f as a parabola in y opening upward The vertex of the parabola occurs with horizontal coordinate 3 + x 5 As f(x 0 0 3) = x 0 0x for x 0 3 we have that f(x 0 y) 0 for y 3 Thus R (G) < 5 (n + ) Case (iii): Assume we have vertices u v x y with d u = d v = 3 u v v y v x and d x d y (a) If d x = and d y 5 then let H denote the graph obtained from G by deleting vertices x v and u Note that H is a connected graph with n 3 3 vertices Thus induction gives R (G) 5 (n ) < 5 (n ) < 5 (n + ) 7

8 (b) Suppose x y If either: d x = d y 4 or d y d x then form a new graph H obtained from G by deleting u and v and adding the edge xy Notice that d H x = d x and d H y = d y Then R (G) = R (H) d x 3d y d x d y If d x = d y = then G is a star on 4 vertices and the inequality holds Otherwise H is a connected graph with n 3 vertices and by induction we have R (G) 5 (n ) d x + 3d y d x d y < 5 (n + ) + d x + d y d x d y 6 6d x d y = 5 (n + ) + ( d x)d y + (d x 3) 6d x d y If d x = d y 4 then the numerator of the second term is nonpositive If d x = or d x = 3 then the the numerator of the second term is negative If d y d x 4 then ( d x )d y + (d x 3) d y + (d y 3) < 0 Hence R (G) < 5 (n + ) holds (c) Suppose x y and d y d x Form a graph H by deleting u and v Note that d H x = d x and d H y = d y Keeping track of the weight of edge xy in G and H gives R (G) < R (H) d x 3d y d x d y (d x )(d y ) Deleting u and v and using induction gives where R (G) < 5 (n + ) f(d x d y ) 6d x d y (d x )(d y ) f(x y) = (x )(x )y (3x 5x 4)y + (x )(x + 3) Our goal is to show that f(x y) 0 for y x (with x y integral) Note that f( y) 0 for y Fix x = x 0 3 and view f as a parabola in y opening upward The vertex occurs with horizontal coordinate 3 + 4x 0 0 (x for x 0 )(x 0 ) 0 = and x 0 3 As f(x 0 3) = x 0 8x for x 0 3 we have that f(x 0 y) 0 for y 3 Thus R (G) < 5 (n + ) Case (iv): Let t and suppose there is a (t s + t)-system of G with s + t 4 Label the vertices as in Figure Then deleting x and y and using induction gives R (G) 5 (n ) + + d r 5 30 (n + ) = 5 (n + ) 8

9 Case (v): Suppose there is a (t s + t)-system of G with s 0 and t 4 Label the vertices as in Figure This system has a subgraph that is a (4 4)-system (that includes the vertices x and y ) By keeping track of the edge weight changes in the (4 4)-system subgraph and deleting x and y we get R (G) 5 (n ) + ( + 4 ) ( ) 3 d r + 3 (d r ) = 5 (n + ) (d r 7)(d r 8) 8d r (d r ) 5 (n + ) since d r is an integer Case (vi): Suppose there is a (k 3 s + k)-system with s + k 4 and k Label the vertices as in Figure This system has a subgraph that is a ( 3 )-system with center R (that includes the vertices x and y) By keeping track of the edge weight changes in the ( 3 )-system and deleting x and y we get R (G) 5 (n ) + ( d R = 5 (n + ) + d R 4 68d R 5 (n + ) ) ( ) 3d R since d R = s + k 4 Case (vii): Suppose there is a (k k + )-system of G for some fixed k Label the vertices as in Figure Let u r j j k be a vertex adjacent to R Form a new graph H obtained from G by deleting the vertices of each ( 3)-system with center r j for j deleting R and adding the edge ur Note that deleting vertex R from G disconnects the graph but by adding the edge ur (and deleting each ( 3)-system with centers r j for j ) we ensure that H is connected The degree of u and r are the same in both G and H Hence R (G) R (H) = 4(k ) 3 + k 3(k + ) + 3(k + ) + d u (k + ) 3d u As we deleted 5(k ) + vertices to form H we have by induction R (G) 5 (n + ) d u(k k + 44) + (k ) 68d u (k + ) < 5 (n + ) since k k + 44 > 0 and k Case (viii): Let k and t [ 3] (a) Suppose there is a (k k + t + )-system of G with center R such that R is also the center of a (t k + t + )-system (note d R = k + t + ) Let u be the vertex adjacent to R that 9

10 is not a vertex of one of the systems with center R Create a new graph H by deleting the vertex R and the vertices of all the systems with center R and adding a ( d u )-system with center vertex u A total of 5(k ) + t + vertices have been deleted Thus we have by induction R (G) 5 (n + ) (5k + t 4)5 + 4k 3 + k 3(k + t + ) + t + t (k + t + ) + d u (k + t + ) 4 3 3d u = 5 (n + ) (k k t + 7kt 34t)d u + (k + t ) 68d u (k + t + ) < 5 (n + ) for t [ 3] and k (b) Suppose G has a (k k + t)-system with center R such that R is also the center of a (t k + t)-system (note d R = k + t) Then n = 5k + t + and every vertex of G belongs to either the (k k + t)-system or the (t k + t)-system Then R (G) = 4k 3 + k 3(k + t) + t + t (k + t) = < 5(n + ) 5(n + ) k + 7kt + 34k + 6t + 6t 68(k + t) Final Case: By Cases (i)-(iii) we may assume that every vertex of degree in G is adjacent to a vertex of degree and further every vertex of degree in G is adjacent to both a vertex of degree and a vertex of degree at least 3 Thus every vertex with degree or is contained in a system of G Note that if G is a (t t)-system then n = t + and R (G) < 5(n+) Thus any (t s + t)-system of G (with s ) must have s s + t 3 and t 3 by Cases (iv) and (v) Any (t s + t)-system with s = belongs to a (k t d)-system of G Any (k t s + k)-system of G must have t 3 by Cases (ii) and (v): t = 3: For (k 3 d)-systems we must have d 5 by Case (vi) Note that if d = k then the graph is a (k 3 k)-system which has R (G) 5(n+) t = : Note that if the graph is a (k k)-system then R (G) < 5 (n + ) If G has a ( )-system then the center of this system has degree forcing G to be a (3 3)-system (which has R (G) < 5 (n + )) Thus for (k s + k)-systems by Case (vii) we must have s Thus in G the center vertex of a (k d)-system and (k 3 d)-system may coincide as with the center vertex of a (k d)-system and a (t d)-system (but not a (k 3 d)-system and (t d)-system) We can partition the vertices of the graph G so as to separate the systems By Case (0) G has at least one system 0

11 Let A be the collection of centers of ( d)-systems with 3 d 3 that do not share a center with any ( d)-system or (k t d)-system Let A be the collection of centers of ( d)-systems with 4 d 3 that do not share a center with any (3 d)-system or (k t d)-system Let A 3 be the collection of centers of (3 d)-systems with 5 d 3 that do not share a center with any (k t d)-system For k let B k be the collection of centers of (k d)-systems with d k + that do not share a center with any (k + d)-system (k 3 d)-system or any (i d)-system for k i For k let C k be the collection of centers of (k 3 d)-systems with d k + that do not share a center with any (k + 3 d)-system or (k d)-system for k For k k let D kk be the collection of centers R such that both a (k d)- system and a (k 3 d)-system have center R but R is not the center of a (k + d)- system or a (k + 3 d)-system For i [ 3] and k [ 3 i] let Ek i be the collection of centers R such that both a (k d)-system and (i d)-system have center R but R is not the center of a (k + d)-system or a (i + d)-system The above sets provide a partition of G into its systems If z is the center of a system of G then either z appears in exactly one set described above or z is the center of a (t t + )- system that belongs to a (k t d)-system (whose center belongs to exactly one set described above) Let Q be the vertices of G that are have degree at least 3 and are not the center of a system of G Then n = Q + 3 A + 5 A + 7 A 3 + k (5k + ) B k + k (7k + ) C k + k k 0 (5k + 7k + ) D kk + (5k + 3) Ek + (5k + 5) Ek + (5k + 7) Ek 3 k= By using (6) we will count the weight on each vertex of G If S is a subset of vertices of G we write w(s) to denote the sum of the weights of the vertices in S Let y Q Then y cannot be adjacent to degree or vertices thus w(y) d y x x y k= 3 = 6 < 5 Let y A and S y be the set of vertices of the ( d y )-system with center y As d y 3 counting the weight on the degree vertex degree vertex and y respectively gives w(s y ) 3 3 k= [ 4 + ( ) + + ( 4 d y d y + d )] y = d y + < 5 3 9d y

12 Let y A and S y be the set of vertices of the ( d y )-system with center y As d y 4 w(s y ) 5 5 [ ( 4 + ( )) + + ( + d )] y = 7d y + 4 < 5 4 d y d y 3 30d y Let y A 3 and S y be the set of vertices of the (3 d y )-system with center y As d y 5 w(s y ) 7 7 [ ( ( )) + + ( 3 4 d y d y + d )] y 3 = 5d y + 3 < 5 3 d y Let y B k and S y be the set of vertices of the (k d y )-system with center y Then w(s y ) 5k + By subtracting 5 [ ( 7 k 5k )) ( + dy + ( k 6 d y 3 + d )] y k 3 from both sides the right hand side factors as w(s y ) 5k + 5 8k 7d y kd y 68(5k + )d y As d y k + we have that 8k 7d y kd y (k 9k + 34) When k = 4 or k = 5 we have k 9k + 34 = 4 Hence w(sy) 5k+ < 5 Let y C k and S y be the set of vertices of the (k 3 d y )-system with center y Then w(s y ) 7k + By subtracting 5 [ ( 7 k 7k ( )) + ( k d y d y 4 + d )] y k 3 from both sides the right hand side factors as w(s y ) 7k + 5 4k 7d y 68(7k + )d y As d y k we have that w(sy) 7k+ < 5 Let y D kk and S y be the set of vertices of the (k d y )-system and (k 3 d y )-system with center y Then [ ( w(s y ) 4 k 5k + 7k + 5k + 7k ) ( 5 + k 6d y 8 + ) 8d y + ( k d y 3 + k 4 + d )] y k k 3 By subtracting 5 As d y k + k we have from both sides the right hand side factors as w(s y ) 5k + 7k + 5 d yk 8k 4k + 7d y 68(5k + 7k + )d y d y k 8k 4k + 7d y k + k k + 3k k

13 But k 5 k so k + k k + 3k k k + 45 > 0 w(s y) 5k +7k + < 5 Hence Fix t [ 3] Let y Ek t and S y be the set of vertices of the (k d y )-system and (t d y )-system with center y Then w(s y ) 5k + t + By subtracting 5 [ ( 4 k 5k + t ) ( + t 6d y + ) + ( dy t + t )] 4d y d y 3 from both sides the right hand side factors as w(s y ) 5k + t + 5 kd y 8k + 6td y t + 7d y 68(5k + t + )d y Since d y = k + t + s with s (by Case (viii)) a simple check verifies that for t [ 3] k [ 3 t] and s [ 3 t k] then kd y 8k + 6td y t + 7d y > 0 w(s Hence y) 5k+t+ < 5 It now follows that R (G) 5 (n + ) by summing the weights on each set of vertices in the partition of G Note that by starting the induction at a higher value of n and using some careful consideration we may improve the bound in Theorem 6 to 5n+C for some constant C < 5 In [9] it is noted there are trees T of every order n such that R (T ) = 5 n + O() Observe that using n 5(n+) 4 instead of then Cases (0)-(iii) in the proof of Theorem 6 hold Thus we can improve the upper bound in the case that G has no suspended paths Observation 7 Let G be a connected graph on n 3 vertices paths then R (G) n 4 If G has no suspended We next look at the effect that edge deletion has on R (G) If G is a graph and e is an edge of G we denote by G e the graph obtained by removing the edge e from G We call an edge e = xy a leaf of G if either d x = or d y = and a non-leaf edge otherwise Note that deleting a leaf edge of G creates an isolated vertex thus in the next two results we assume the edge being deleted is a non-leaf edge Lemma 8 [6 Lemma 33] Let G be a graph and let e be an edge whose weight is minimal over all edges in G If e is a non-leaf edge then R (G e) > R (G) In the next theorem we determine the maximum change that can occur when deleting an edge 3

14 Theorem 9 Let G be a graph and let e be a non-leaf edge of G then Furthermore if G e is connected then R (G) 4 < R (G e) R (G) R (G e) R (G) Proof Let e = uv and d u denote d G u and d v denote d G v As e is a non-leaf edge we have d u d v Then R (G) R (G e) = d u d v d u (d u ) d i d v (d v ) d i i v i u i u i v Thus R (G) R (G e) < d u d v 4 which gives the first inequality Similarly as d i R (G) R (G e) d u d v d u d v It is not too hard to see that over the integers and for d u d v the right hand side is minimal when d u = d v = Hence R (G) R (G e) 3 4 If G e is connected then there are vertices î v ĵ u (with possibly î = ĵ) such that î u ĵ v dî > and dĵ > Thus R (G) R (G e) d u d v d u (d u ) d u d u (d u ) d v (d v ) d v d v (d v ) It is not too hard to see that over the integers and for d u d v the right hand side is minimal when d u = d v = 3 Hence in the case that G e is connected R (G) R (G e) 7 8 We illustrate the sharpness of Theorem 9 with three examples Let G be the path on 4 vertices which has R (G) = 5 Removing the non-leaf edge e of G gives a disconnected graph with R (G e) = Thus in this case R (G e) = R (G) Let Ĝ be the path x x x 7 on 7 vertices and add the edge e = x x 6 to form a graph G Then Ĝ = G e is connected and R (G e) = R (G) Let G be the graph of order n composed of a K n with a triangle xyz attached to a vertex z of the K n Then using the edge e = xy we have R (G) R (G e) = 4 n By taking n the right hand side can be made arbitrarily close to 4 4

15 3 Bounds on the L-energy of a graph Recall that the L-energy of a graph G is E L (G) = n λ i (L) i= Using Lemma along with the results in Section bounds can be derived on the L-energy of a graph If G has k connected components in particular G G G k then E L (G) = k E L (G i ) (7) We first provide a bound on the L-energy of a graph with k connected components i= Lemma 0 Let G be a graph of order n with k connected components and no isolated vertices Then E L (G) k + (n k)(r (G) k) Proof Note that is an eigenvalue of I L with multiplicity k hence n k E L (G) = k + λ i (I L) i= By the Cauchy-Schwartz inequality (using vectors ( ) T and ( λ (I L) λ n k (I L) ) T ) we obtain the upper bound n k E L (G) k + (n k) [λ i (I L)] The result now follows by (5) We next provide bounds on the L-energy in terms of the minimum and maximum degrees of G Corollary Let G be a graph of order n with k connected components and no isolated vertices Suppose G has minimum vertex degree equal to d min and maximum vertex degree equal to d max Then Furthermore n n n d max E L (G) E L (G) k i= n dmin n Proof Lemma and Theorem gives the first string of inequalities For the last inequality by (7) it suffices to prove E L (G) k in the case that k = Note that λ n (I L) = and the trace of I L is 0 Thus n E L (G) = + λ i (I L) + i= 5 n λ i (I L) = + = (8) i=

16 Note that if G is a regular graph of degree r then n r E L(G) n r Over the graphs of order n with no isolated vertices we characterize those that have maximal and minimal L-energy Corollary Let G be a graph of order n with no isolated vertices Then E L (G) with equality if and only if G is a complete multipartite graph Further with equality only for the following cases: E L (G) n/ (i) n is even and G is the disjoint union of n/ paths of length or (ii) n is odd and G is the disjoint union of (n 3)/ paths of length and one path of length or (iii) n is odd and G is the disjoint union of (n 3)/ paths of length and a complete graph on 3 vertices Proof Equality in (8) occurs if and only if λ n (I L) 0 (equivalently λ n (A) 0) It is known that the adjacency matrix of G has only one positive eigenvalue if and only if G is a complete multipartite graph plus isolated vertices (see [5]) By Theorem we have E L (G) n It can be seen that for equality to hold we must have R (G) = n/ and G must be regular of degree Thus G is the disjoint union of n/ paths of length which indeed has E L (G) = n Note that both the path of length and the complete graph on 3 vertices have energy Hence if n is odd the graphs described in (ii) and (iii) have energy n It is easy to see that if n is odd and E L (G) = n then any even connected component of G must have size If there is an odd connected component Ĝ of G of size k 7 then by Lemma and Theorem 6 E L (Ĝ) < k If there is a connected component Ĝ of order 5 and if Ĝ has no suspended paths then by Lemma and Observation 7 E L (Ĝ) < 4 If Ĝ is of order 5 and has a suspended path then there are only three such graphs and each has E L (Ĝ) < 4 Hence any odd connected component must be of order 3 and since E L (G) = n there can only be one such odd connected component The upper bound in Corollary can be improved for connected graphs by using Lemma and Theorem 6 Corollary 3 If G is a connected graph on n 3 vertices then 5 E L (G) < (n + ) < 073(n + ) 8 Furthermore if G has no suspended paths (or more generally R (G) n 4 ) then E L (G) n < 0707 n 6

17 One might suspect that over the connected graphs that the path has maximal L-energy but in general this is not true We next provide some common classes of graphs along with their corresponding L-energy Example 4 Let G be a path on n vertices Using the eigenvalues of L (see [3]) we get that (n )/ E L (G) = cos(kπ/(n )) By [7 page 37] k=0 Thus for the path E L (G) π n k=0 N ( ) ( ) Nx N + ( x ) cos(kx) = cos sin x csc Example 5 (a) For n odd let G be a (t t)-system with n = t+ vertices The normalized Laplacian matrix of G can be written in block form as I t I t 0 L = I t I t t 0 T t T where I t represents the identity matrix of order t 0 represents the 0 vector of size t and represents the all ones vector of size t Thus the eigenvalues are 0 each with multiplicity and ( ± ) each with multiplicity t Hence the L-energy is E L (G) = n 3 + n (b) For n = t + even let G be the graph obtained by joining a vertex to a leaf of a (t t)-system The normalized Laplacian eigenvalues of this graph are 0 and each with multiplicity ( ± ) each with multiplicity t along with four other eigenvalues This is enough to obtain E L (G) n It should be noted that if G is a (t 3 t)-system with n = 7t+ then using a computer to test large values of n suggests that E L (G) 067n Similarly if G is a (t t)-system with n = 5t + then using a computer to test large values of n suggests that E L (G) 0648n These values are far from the upper bound given by Corollary 3 For n = 3 the path and triangle each have maximal L-energy For 4 n 6 the path has maximal L-energy over the class of connected graphs Note that for 4 n 6 the path falls under the class of graphs described in Example 5 For 7 n 8 a computer has verified that over all connected graphs the class of graphs in Example 5 have maximal L-energy For n 9 it is unknown which graphs have maximal L-energy We know of no class of connected graphs on n vertices that has L-energy (asymptotically) larger than n Corollary 3 implies such a graph G would have R (G) > n 4 and 7

18 Observation 7 suggests such a graph should have a large number of suspended paths We ask the question: Over the connected graphs G of order n is E L (G) n + C for some suitable constant C? We now look at other bounds on L-energy Theorem 6 Let G be a graph of order n with no isolated vertices and let = det(i L) Then E L (G) R (G) + n(n ) /n Proof For convenience we use λ i to denote λ i (L) Note that E L (G) = R (G) + i j λ i λ j By the arithmetic-geometric mean inequality n(n ) λ i λ j λ i λ j i j i j n(n ) = /n Hence the result now follows We next relate the L-energy of a graph G to its A-energy where A is the adjacency matrix of G Recall that the A-energy is simply E A (G) = n λ i (A) This quantity has been well studied by a large number of authors (see for example [9]) i= Theorem 7 Let G be a graph of order n with no isolated vertices Suppose d min and d max are the minimum and maximum vertex degrees of G respectively Then d min E L (G) E A (G) d max E L (G) Proof The proof uses a Theorem due to Ostrowski [ Theorem 459] As D / is nonsingular for each k = n there is a positive real number θ k such that and Thus from which the result now follows = λ (D ) θ k λ n (D ) = d max d min λ k (I L) = θ k λ k (A) E L (G) = n θ k λ k (A) k= 8

19 Theorem 7 implies that if G is a regular graph of degree r then E A (G) = re L (G) Since E A (G) is well studied many bounds for E A (G) in the literature can be applied to E L (G) by way of Theorem 7 We now look at the effect edge deletion has on E L (G) We begin with examples to show that L-energy can increase decrease or remain unchanged upon edge deletion The examples will also illustrate that the effect edge deletion has on the general Randić index does not necessarily provide direct information about the effect edge deletion has on L-energy Example 8 In this example we list the L-energy and Randić index (to three decimal places if appropriate) for each graph in Figures 3 4 and 5 (i) The graphs in Figure 3 have a decrease in L-energy upon deleting edge e For the first (resp second and third) graph = E L (G e) < E L (G) 457 (resp = E L (G e) < E L (G) 68 and = E L (G e) < E L (G) 704) For the first (resp second and third) graph = R (G e) > R (G) 097 (resp R (G e) = R (G) = and = R (G e) < R (G) = 05) (ii) The graphs in Figure 4 have an increase in L-energy upon deleting edge e For the first (resp second and third) graph 869 E L (G e) > E L (G) 667 (resp 3076 E L (G e) > E L (G) 904 and 37 E L (G e) > E L (G) = 3) For the first (resp second and third) graph R (G e) > R (G) 08 (resp R (G e) = R (G) 007 and 098 R (G e) < R (G) 0978) (iii) The graphs in Figure 5 have no change in L-energy upon deleting edge e For the first (resp second) graph E L (G e) = E L (G) = (resp E L (G e) = E L (G) 78) For the first (resp second) graph = R (G e) > R (G) = 075 (resp R (G e) = R (G) = 065) We are not aware of a graph G where upon edge deletion L-energy remains constant while R (G) decreases e e e Figure 3: L-energy decreases upon deleting edge e e e e Figure 4: L-energy increases upon deleting edge e The next result provides a bound on how much the L-energy can change upon edge deletion Theorem 9 Let G be a graph of order n without isolated vertices and let e be a non-leaf edge of G Then 3 E L (G) E L (G e)

20 e e Figure 5: L-energy remains constant upon deleting edge e Proof Let L G and L G e be the normalized Laplacian matrices of G and G e and suppose e = xy Let C = L G L G e Observe that by [6] (namely σ i (A + B) σ i (A) + σi (B)) we can derive Note that rank(c) 4 λ λ λ 3 λ 4 Then E L (G) E L (G e) n σ i (C) i= Let the eigenvalues of C be 0 with multiplicity n 4 and n σ i (C) = λ + λ + λ 3 + λ 4 i= By the Cauchy-Schwartz inequality E L (G) E L (G e) tr(c ) which equals: ( ) + ( dx d y j y j x dx d j ) + ( (dx )d j j x j y dy d j ( 4 + dx ) ( d x dy ) d y + d x d y (dy )d j ) The 4 comes from setting d x = d y = as this is when the first term is maximal and the other two expressions come from noticing d j The function ( ) x x f(x) = x has f 3 (x) < 0 for x > Thus as d x d y E L (G) E L (G e) 4 References [] S Akbari E Ghorbani and MR Oboudi Edge addition singular values and energy of graphs and matrices Linear Algebra and its Applications 430 (009) 9-99 [] B Bollobás and P Erdős Graphs of extremal weights Ars Combin 50 (998) 5-33 [3] FRK Chung Spectral Graph Theory American Math Soc Providence 997 0

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