Connectivity and minimal distance spectral radius of graphs

arXiv:1011.2049v1 [math.SP] 9 Nov 2010 Connectivity and Minimal Distance Spectral Radius of Graphs Xiaoling Zhangab ∗ †, Chris Godsila∗ a Combinatorics and Optimisation University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 b School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China Abstract In this paper, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with k cut vertices (respectively, k cut edges) with the minimal distance spectral radius. Key words: Distance spectral radius; Pendant path AMS subject classification: 05C50; 15A18 1 Introduction The distance matrix of a graph, while not as common as the more familiar adjacency matrix, has nevertheless come up in several different areas, in∗ E-mail addresses: [email protected], [email protected]. The author is currently a visiting Ph.D. student at the Department of Combinatorics and Optimisation in University of Waterloo from September 2008 to September 2009. She is partially supported by NSFC grant no. 10831001. † 1 cluding communication network design [6], graph embedding theory [4, 7, 8], molecular stability [10, 11], and network flow algorithms [3, 5]. So it is interesting to study the spectra of these matrices. In this paper, we study the largest eigenvalue of the distance matrix of a graph. Throughout this paper, we will assume that G is a simple, connected graph of order n, that is, with n vertices. Let G be a connected graph with vertex set {1, . . . , n}. The distance between vertices i and j of G, denoted by dist(i, j), is defined to be the length (i.e., the number of edges) of the shortest path from i to j [2]. The distance matrix of G, denoted by D(G) is the n × n matrix with its (i, j)-entry equal to dist(i, j), i, j = 1, 2, . . . , n. Note that dist(i, i) = 0, i = 1, 2, . . . , n. The distance eigenvalue of largest magnitude is called the distance spectral radius, and is denoted by Λ1 . Balaban et al. [1] proposed the use of Λ1 as a structure-descriptor, and it was successfully used to make inferences about the extent of branching and boiling points of alkanes[1, 9]. In this paper, we determine the graph with k cut vertices (respectively, k cut edges) which has the minimal distance spectral radius. Main results Let G be a connected graph. Let deg(v) (or degG (u)) denote the degree of the vertex v in G. We define a pendant path of G to be a walk v0 v1 · · · vs (s > 1) such that the vertices v0 , v1 , . . . , vs are distinct, deg(v0 ) > 2, deg(vs ) = 1, and deg(vi ) = 2, whenever 0 < i < s. And v0 , s are called the root and the length of the pendant path, respectively. We give a generalization of Theorem 3.5 in [12]. Theorem 1.1. Let u and v be two adjacent vertices of a connected graph G and for positive integers k and l, let Gk,l denote the graph obtained from G by adding paths of length k at u and length l at v. If k > l > 1, then Λ1 (Gk,l ) < Λ1 (Gk+1,l−1); if k = l > 1, then Λ1 (Gk,l ) < Λ1 (Gk+1,l−1 ) or Λ1 (Gk,l ) < Λ1 (Gk−1,l+1 ). 2 1 1 1 k +1 k +1 k +2 k +2 k +4 k +3 l +k +2 Gk ,l k k +1 k +2 k +3 l+k +2 Gk +1,l - 1 l+k +2 Gk - 1,l +1 Fig. 1. Graphs Gk,l , Gk+1,l−1, Gk−1,l+1. Proof: Label vertices of Gk,l , Gk+1,l−1, Gk−1,l+1 as in Figure 1. We partition V (Gk,l ) into A ∪ {k + 2} ∪ B ∪ C ∪ D ∪ E, where A = {1, . . . , k + 1}, B = {k + 3, . . . , l + k + 2}, C = {j | dist(j, k + 1) + 1 = dist(j, k + 2)}, D = {j | dist(j, k + 1) = dist(j, k + 2)}, E = {j | dist(j, k + 1) = dist(j, k + 2) + 1}. When we pass from G to G′ , the distances within A ∪ {k + 2} ∪ B and within C ∪ D ∪ E are unchanged; the distances between A and C ∪ D ∪ E , {k + 2} and E are increased by 1; the distances between B and C ∪D∪E, {k +2} and C are decreased by 1; the distances between {k + 2} and D are unchanged. If the distance matrices are partitioned according to A, {k + 2}, B, C, D, E, their difference is 3      D(Gk+1,l−1)−D(Gk,l ) =      0 0 0 eA (eC )t eA (eD )t eA (eE )t 0 0 0 −(eC )t 0 (eE )t 0 0 0 −eC (eC )t −eC (eD )t −eC (eE )t (eC )t eA −eC −(eC )t eC 0 0 0 t t (eD ) eA 0 −(eD ) eC 0 0 0 (eE )t eA eE −(eE )t eC 0 0 0 where ei = (1, . . . , 1)t | {z } |i| and i = A, C, D, E. Let x = (x1 , . . . , xn )t be a positive eigenvector corresponding to Λ1 (Gk,l ). Then we have

1 t 1 Λ1 (Gk+1,l−1) − Λ1 (Gk,l ) > x D(Gk+1,l−1) − D(Gk,l) x (1.1) 2 2 ! X X X = xj − xj xj j∈A j∈B −xk+2 X j∈C∪D∪E xj + xk+2 j∈C > X xj − j∈A Similarly, we have X xj j∈E X xj − xk+2 j∈B ! X j∈C∪D∪E

1 1 t Λ1 (Gk−1,l+1 ) − Λ1 (Gk,l ) > x D(Gk−1,l+1)) − D(Gk,l) x 2 2 ! X X = xj + xk+2 − xj + xk+1 j∈B j∈A +xk+1 X xj − xk−2 j∈C > X Since D(Gk,l )x = Λ1 (Gk,l )x, i.e., Λ1 (Gk,l )xi = X j∈A n P j=1 n, we get: 4 X xj j∈C∪D∪E xj j∈E xj + xk+2 − j∈B X xj . xj ! X xj . j∈C∪D∪E dist(i, j)xj , for 1 6 i 6      ,     if 1 6 i 6 l + 1, then Λ1 (Gk,l )(xi − xl+k+3−i ) = s X m=1 X (l + k − 2m + 3)(xl+k+3−m − xm ) + (k − l) (1.2) xj j∈D +(k − l − 1) X xj + (k − l + 1) j∈C −2 i−1 X X xj j∈E (xl+k+3−m − xm ); m=1 if l + 2 6 i 6 s, then Λ1 (Gk,l )(xi − xl+k+3−i ) = s X (l + k − 2m + 3)(xl+k+3−m − xm ) (1.3) m=1 +(k + l + 3 − 2i) X j∈C∪D∪E where s = l+k+2 , if l + k is even; and s = 2 From Eqs. (1.3) and (1.4), we get: if 2 6 i 6 l + 1, then l+k+1 , 2 xj − 2 i−1 X (xl+k+3−m − xm ); m=1 if l + k is odd. Λ1 (Gk,l )(xi−1 − xl+k−i+4 ) − Λ1 (Gk,l )(xi − xl+k+3−i ) i−1 X =2 (xl+k+3−r − xr ); (1.4) Λ1 (Gk,l )(xi−1 − xl+k−i+4 ) − Λ1 (Gk,l )(xi − xl+k+3−i ) i−1 X X X =2 (xl+k+3−r − xr ) + xj + 2 xj ; (1.5) Λ1 (Gk,l )(xi−1 − xl+k−i+4 ) − Λ1 (Gk,l )(xi − xl+k+3−i ) i−1 X X =2 (xl+k+3−r − xr ) + 2 xj . (1.6) r=1 if i = l + 2, then r=1 j∈D j∈E if l + 3 6 i 6 s, then r=1 j∈C∪D∪E Equation (1.5) implies that, for 1 6 i 6 l + 1, the differences xl+k+3−i − xi are either all non-positive or non-negative. 5 Case 1. k = l. If xl+k+3−i − xi 6 0, for all 1 6 i 6 l + 1, then X X xj − xj − xk+2 > 0. j∈A j∈B So we get Λ1 (Gk,l ) 6 Λ1 (Gk+1,l−1). Here, if Λ1 (Gk,l ) = Λ1 (Gk+1,l−1 ), then the last equality in (1.2) holds, which implies that X X xj + xj = 0, 2 j∈E j∈D i.e., D = E = ∅. Since i−1 6 l = s −1, for 1 6 m 6 i−1, l + k −2m+ 3 > 2, we get s X (l + k − 2m + 3)(xl+k+3−m − xm ) − 2 m=1 i−1 X (xl+k+3−m − xm ) 6 0. m=1 Combining this with the fact that x is a positive eigenvector corresponding to Λ1 (Gk,l ), we get that the right side of equation (1.3) is strictly less than 0, which contradicts the fact that the left side of equation (1.3) Λ1 (Gk,l )(xi − xl+k+3−i ) > 0. So we get Λ1 (Gk,l ) < Λ1 (Gk+1,l−1). If xl+k+3−i − xi > 0, for all 1 6 i 6 l + 1, we can get Λ1 (Gk,l ) < Λ1 (Gk−1,l+1) similarly. Case 2. k > l. If xl+k+3−i − xi > 0, for all 1 6 i 6 l + 1, then from Eqs. (1.6) and (1.7), we can get that xl+k+3−i − xi > 0, for all l + 2 6 i 6 s. Similar to the above case, we get that the right side of equation (1.5) is strict larger than 0, which contradicts the fact that the left side of equation (1.3) Λ1 (Gk,l )(xi − xl+k+3−i ) 6 0. So this case is impossible. P P If xl+k+3−i − xi < 0, then xj − xj − xk+2 > 0, which implies that j∈A j∈B Λ1 (Gk,l ) > Λ1 (Gk+1,l−1). This completes the proof. From the proof of above theorem, we get the following corollary. 6 2 Corollary 1.2. Let vl and vm be two adjacent vertices of connected graph G. Let Pl and Pm be two pendant paths with roots vl and vm , respectively. If l > m, then X X xj > xj . j∈V (Pl ) j∈V (Pm ) Theorem 1.3. Let C1 be a component of G − u and v1 , . . . , vk (1 6 k 6 degG (u) − degC1 (u)) be some vertices of NG (u) \ NC1 (u). Suppose NC1 (u) \ {v} = NC1 (v), where v is a vertex of C1 adjacent to u. Let G′ be the graph obtained from G by deleting the edges uvs and adding the edges vvs (1 6 s 6 k). If there exists a vertex w ∈ V (G) \ (V (C1 ) ∪ {u}) such that distG (w, vs ) < distG′ (w, vs ), for all 1 6 s 6 k, then Λ1 (G) < Λ1 (G′ ). Proof: We partition V (G) into A ∪ B ∪ {u} ∪ {v} ∪ C, where A = {v1 , . . . , vk }, B = V (G) \ (A ∪ V (C1 ) ∪ {u}), C = V (C1 ) \ {v}. From G to G′ , the distances between A ∪ B and C are unchanged; the distances between A and {u} ∪ {w} are increased by 1; the distances between A and {v} are decreased by 1; the distances between A and B \ {w} are not decreased. Let x = (x1 , . . . , xn )t be a positive eigenvector corresponding to Λ1 (G). Similar to the proof of above theorem, we have

1 1 t Λ1 (G′ ) − Λ1 (G) > x D(G′ ) − D(G) x 2 2 X > (xw + xu − xv ) xj . j∈A 7 Since D(G)x = Λ1 (G)x, i.e., Λ1 (G)xi = n P dist(i, j)xj , we can easily get j=1 (Λ1 (G) + 1)(xw + xu − xv ) = n X (dist(w, j) + dist(u, j) − dist(v, j)) xj + (xw + xu − xv ) j=1 = X (dist(w, j) + dist(u, j) − dist(v, j))xj j∈A∪(B\{w}) + X (dist(w, j) + dist(u, j) − dist(v, j))xj j∈C +(dist(w, w) + dist(u, w) − dist(v, w))xw +(dist(w, u) + dist(u, u) − dist(v, u))xu +(dist(w, v) + dist(u, v) − dist(v, v))xv + (xw + xu − xv ). As we know, dist(u, j) − dist(v, j) = −1 and dist(w, j) > 1, for j ∈ A ∪ (B \ {w}), so X (dist(w, j) + dist(u, j) − dist(v, j))xj > 0. j∈A∪(B\{w}) Similarly, we can get X (dist(w, j) + dist(u, j) − dist(v, j))xj > 0. j∈C Since dist(w, w) + dist(u, w) − dist(v, w) = −1, dist(w, u) + dist(u, u) − dist(v, u) > 0, dist(w, v) + dist(u, v) − dist(v, v) > 3, combining these with all the above equations and inequations, we get
Λ1 (G) + 1)(xw + xu − xv > 0, which implies Λ1 (G′ ) > Λ1 (G). This completes the proof. 2 8 2 Applications The graph Gn,k is a graph obtained by adding paths Pl1 +1 , . . . , Pln−k +1 of almost equal lengths (by the length of a path, we mean the number of its vertices) to the vertices of the complete graph Kn−k ; that is, the lengths l1 , . . . , ln−k of Pl1 +1 , . . . , Pln−k +1 which satisfy |li − lj | 6 1; 1 6 i, j 6 n − k. Knk is a graph obtained by joining k independant vertices to one vertex of Kn−k . Theorem 2.1. Of all the connected graphs with n vertices and k cut vertices, the minimal distance spectral radius is obtained uniquely at Gn,k . Proof: We are supposed to prove that if G is a connected graph with n vertices and k cut vertices, then Λ1 (G) > Λ1 (Gn,k ) with equality only when G∼ = Gn,k . Let V1 be the set of the cut vertices of G. Note that if we add some edges to G such that each block of G − V1 is a clique, denoting the new graph by G′ , then distG (i, j) > distG′ (i, j). Consequently, D(G) > D(G′ ), which implies Λ1 (D(G)) > Λ1 (D(G′ )) with equality only when D(G) = D(G′ ). So, in the following, we always assume that each cut vertex of G connects exactly two blocks and that all these blocks are cliques. Order the cardinalities of these blocks a1 > a2 > · · · > ak+1 > 2 and denote the blocks by Ka1 , . . . , Kak+1 . If k = 0, then G ∼ = Kn and the theorem holds. If k = n − 2, then G is the path Gn,n−2. If k = n−3, then a1 = 3, a2 = · · · = ak+1 = 2. The result follows from a repeated use of Theorem 1.1. Thus we may assume that 1 6 k 6 n−4. Moreover, we observe that a1 = n+k−(a2 +· · ·+ak+1 ) 6 n−k. Choose G such that the distance spectral radius is as small as possible. Claim. a1 = n − k. Otherwise, a1 6 n − k − 1, which implies a2 > · · · > ai > 3 for some i, 2 6 i 6 k + 1. Suppose Kai1 , . . . , Kait are the blocks, each of which contains at least two roots of pendant paths of G. Let P be the set of pendant paths whose roots are contained in Kai1 , . . . , Kait and Pm be one of the shortest pendant paths among P. Suppose the root of Pm is contained in Kais for some 1 6 s 6 t 9 and Pl is another pendant path whose root is also contained in Kais . Then we have 0 6 l − m 6 1. Otherwise, by Theorem 1.1, we can find a graph G′ such that Λ1 (G) > Λ1 (G′ ), which is a contradiction. We label the vertices of G such that Pm = v1 · · · vm , Pl = vm+1 · · · vm+l , V (Kais ) \ {vm , vm+1 } = {u1 , . . . , ur }, where vm and vm+1 are the roots of Pm and Pl , respectively. Suppose u1 is a cut vertex of G such that G[V (C1 ) ∪ {u1 }] contains at least one block Kah , for some 1 6 h 6 i, where C1 is the component of G − u1 which does not contain vm . Let NC1 (u1 ) = {w1 . . . . , wq } and G′ = G − vm u2 − · · · − vm ur − vm vm+1 + w1 u2 + · · · + w1 ur + w1 vm+1 + ········· + wq−1 u2 + · · · + wq−1 ur + wq−1 vm+1 + wq u2 + · · · + wq ur + wq vm+1 . Then G′ is a connected graph with n vertices and k cut vertices. In the following, we consider the graph G′ . Since V (G′ ) = V (G), we partition V (G′ ) into A ∪ {u1 } ∪ B ∪ C, where A = V (Pm ), B = V (C1 ), C = V (G) \ (A ∪ {u1 } ∪ B). From G′ to G, the distances between A and B ∪ {u1 }, {u1 } and B are unchanged; the distances between B and C are increased by 1; the distances between A and C are decreased by 1. Let x = (x1 , . . . , xn )t be a positive 10 eigenvector corresponding to Λ1 (G′ ). Then we have

1 1 t Λ1 (G) − Λ1 (G′ ) > x D(G) − D(G′ ) x 2 2 ! X X X > xj − xj xj . j∈B Case 1. P xj − j∈B P j∈A (2.1) j∈C xj > 0. j∈A From inequality (2.2), we get that Λ1 (G) > Λ1 (G′ ), which is a contradiction. P P Case 2. xj − xj 6 0 j∈B j∈A Since D(G′ )x = Λ1 (G′ )x, i.e., Λ1 (G′ )xi = n P dist(i, j)xj , we can easily j=1 get that Λ1 (G′ ) X xj − j∈B X j∈A xj ! X = i∈A∪{u1 } + dist(u, i) − u∈B X dist(u, i) − ! dist(v, i) (2.2) xi v∈A X v∈A u∈B X X i∈C dist(u, i) − u∈B X X i∈B + X X v∈A ! dist(v, i) xi ! dist(v, i) xi . Since G[V (C1 )∪{u1 }] = G′ [V (C1 )∪{u1 }] contains at least one block Kah , for some 1 6 h 6 i, G′ [V (C1 ) ∪ {u1 }] must contain at least two pendant paths P ′ and P ′′ whose roots are contained in the same block. Denote the roots of P ′ and P ′′ by ω1 and ω2 , respectively. Suppose distG′ (vm+1 , ω1 ) = k, then k > 2 and distG′ (vm+1 , ω2 ) = k. As we know, P ′ and P ′′ are two pendant paths of length at least m, so if i ∈ A ∪ {u1 }, then X u∈B dist(u, i) − X v∈A dist(v, i) > 2 × (m + k − 1)(m + k) m(m − 1) − ; (2.3) 2 2 11 if i ∈ B, then X dist(u, i) − u∈B X dist(v, i) > v∈A (k − 2)(k − 1) − (m − 1)k; 2 (2.4) if i ∈ C, then X dist(u, i) − u∈B X dist(v, i) > 0. (2.5) v∈A is obvious. Combining Eqs. (2.3), (2.3), (2.4) with (2.5), we get ! X X Λ1 (G′ ) xj − xj j∈B j∈A   X (m + k − 1)(m + k) m(m − 1) > 2× xi − 2 2 i∈A∪{u1 } X  (k − 2)(k − 1) − (m − 1)k xi + 2 i∈B  X m(m − 1) (k − 2)(k − 1) > (m + k − 1)(m + k) − + − (m − 1)k xi 2 2 i∈B > 0, which is a contradiction. So this case does not exist. Up to now, we have proved the claim that a1 = n − k, which implies that a2 = · · · = ak+1 = 2. From a repeated use of Theorem 1.1, we get that G ∼ = Gn,k . This completes the proof. 2 Theorem 2.2. Of all the connected graphs with n (n > 4) vertices and k cut edges, the minimal distance spectral radius is obtained uniquely at Knk . Proof: We are supposed to prove that if G is a connected graph with n vertices and k cut edges, then Λ1 (G) > Λ1 (Knk ) with equality only when 12 G∼ = Knk . Let E1 = {e1 , e2 , . . . , ek } be the set of the cut edges of G. For a similar reason to Theorem 2.1, we assume that each component of G − E1 is a clique. If k = 0, then G ∼ = Kn and the theorem holds. So we assume that k > 1. Denote the components of G − E1 by Ka0 , . . . , Kak , where a0 + · · · + ak = n. Let Vai = {v ∈ Kai : v is an end vertex of the cut edges of G}, and choose G such that the distance spectral radius is as small as possible. Claim 1. |Vai | = 1, 0 6 i 6 k. Otherwise, |Vai | > 1 for some 0 6 i 6 k. Suppose u, v ∈ Vai and vvj , uvh ∈ E1 . Let G′ = G − vvj + uvj . Then, G′ is still a connected graph with n vertices and k cut edges, and distG′ (vh , vj ) < distG (vh , vj ). In G′ , NKai (u) \ {v} = NKai (v) \ {u}, so by Theorem 1.3, we get Λ1 (G′ ) < Λ1 (G), which is a contradiction. So, in the following, we can assume that Vai = {vi }, 0 6 i 6 k. Claim 2. If vs is adjacent to vt , where 0 6 t, s 6 k, then deg(vs ) = 1 or deg(vt ) = 1. Otherwise, deg(vs ) > 2 and deg(vt ) > 2. Denote by C1 , C2 the components of G−vs vt which contain vs and vt , respectively. Suppose N(vs )\{vt } = {vk+1 , . . . , vk+r } and u ∈ N(vt ) \ {vs }. Let G′ = G − vs vk+1 − . . . − vs vk+r + vt vk+1 + . . . + vt vk+r . Then, G′ is still a connected graph with n vertices and k cut edges, and distG′ (u, vk+i ) < distG (u, vk+i ) (1 6 i 6 r). In G′ , vs vt is a pendant edge, so by Theorem 1.3, we get Λ1 (G′ ) < Λ1 (G), which is a contradiction. Since G is connected, combining Claim 1 with Claim 2, we get that G ∼ = k Kn . This completes the proof. 2 13 References [1] A. T. Balaban, D. Ciubotariu, M. Medeleanu, Topological indices and real number vertex invariants based on graph eigenvalues or eigenvectors. J. Chem. Inf. Comput. Sci. 31 (1991) 517-523. [2] F. Buckley, F. 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