The inertia and energy of the distance matrix of a connected graph

Linear Algebra and its Applications 467 (2015) 29–39

Contents lists available at ScienceDirect

Linear Algebra and its Applications www.elsevier.com/locate/laa

The inertia and energy of the distance matrix of a connected graph ✩ Huiqiu Lin a,∗ , Ruifang Liu b , Xiwen Lu a a

Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China b School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China

a r t i c l e

i n f o

Article history: Received 25 June 2014 Accepted 31 October 2014 Available online 19 November 2014 Submitted by R. Brualdi MSC: 05C50 Keywords: D-eigenvalue The distance spectral radius Inertia Distance energy

a b s t r a c t Let G be a connected graph and D(G) be the distance matrix of G. Suppose that λ1 (D) ≥ λ2 (D) ≥ · · · ≥ λn (D) are the D-eigenvalues of G. In this paper, we show that the distance matrix of a clique tree is non-singular. Moreover, we also prove that the distance matrix of a clique tree has exactly one positive D-eigenvalue. In addition, we determine the extremal graphs with maximum and minimum distance energy among all clique trees. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Unless stated otherwise, we follow [3] for terminology and notations, and we consider finite connected simple graphs. In particular, denote by V (G) = {v1 , . . . , vn } the vertex set of G, E(G) the edge set of G. For a graph G = (V, E), two vertices are called adjacent ✩ Supported by the National Natural Science Foundation of China (Nos. 11371137, 11401211 and 11201432). * Corresponding author. E-mail addresses: [email protected] (H. Lin), rfl[email protected] (R. Liu), [email protected] (X. Lu).

http://dx.doi.org/10.1016/j.laa.2014.10.045 0024-3795/© 2014 Elsevier Inc. All rights reserved.

30

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

Fig. 1. A clique tree and a clique path.

if they are connected by an edge and two edges are called incident if they share a common vertex. We call |V (G)| the order of G and |E(G)| the size of G. As usual, let Kn , K1,n−1 , and Pn denote the complete graph, the star, and the path with order n, respectively. The wheel graph with n vertices Wn is the graph that contains a cycle of length n − 1 plus a vertex v not in the cycle such that v is connected to every other vertex. The line graph of G, denoted as L(G), is the graph whose vertex set is E(G); two vertices of L(G) are adjacent if the corresponding edges of G are incident. A block of G is a maximal connected subgraph of G that has no cut-vertex. A graph G is a clique tree if each block of G is a clique. We call Pn1 ,...,nk a clique path if we replace each edge of Pk+1 by a clique Kni such that V (Kni ) ∩ V (Kni+1 ) = vi for i = 1, . . . , k − 2 and V (Kni ) ∩ V (Knj ) = ∅ for j = i − 1, i + 1 and 2 ≤ i ≤ k − 1. We call Ku,n1 ,...,nk a clique star if we replace each edge of the star K1,k with a clique Kni such that V (Kni ) ∩ V (Knj ) = u for i = j and i, j = 1, . . . , k (see Fig. 1). Obviously, both a tree T and the line graph of T are clique trees. Let G be a connected graph. The matrix D(G) is nonnegative and irreducible, so the eigenvalues of D(G) are real and we can order the eigenvalues as λ1 (D) ≥ λ2 (D) ≥ · · · ≥ λn (D). The largest eigenvalue of D(G) is called the distance spectral radius of G, denoted by λ1 (D). The positive unit eigenvector corresponding to λ1 (D) is called the Perron vector of D(G). There are a lot of papers about the distance spectral radius, see [9,10,14,15]. The inertia of the matrix M is the triple of integers (n+ (M ), n0 (M ), n− (M )), where n+ (M ), n0 (M ) and n− (M ) denote the number of positive, 0 and negative eigenvalues of M , respectively. If det(M ) = 0, then we call M singular; otherwise, we call M nonsingular. I. Gutman and I. Sciriha [8] showed that n0 (A(L(T ))) ≤ 1 where A(L(T )) denotes the adjacency matrix of the line graph of a tree T . Motivated by their result, in this paper, we show that n0 (D(L(T ))) = 0 where D(L(T )) denotes the distance matrix of the line graph of a tree T , that is, D(L(T )) is non-singular. Problem 1.1. For which connected graph G, does D(G) contain exactly one positive D-eigenvalue? Up to now, only a few graphs are known to have exactly one positive D-eigenvalue, such as trees [7], connected unicyclic graphs [2], the polyacenes, honeycomb and square lattices [1], complete bipartite graphs [6], Kn , and iterated line graphs of some regular

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

31

graphs [16]. Recently, X. Zhang and C. Godsil [20] not only proved that cacti have exactly one positive D-eigenvalue but also gave a construction for graphs which have exactly one positive D-eigenvalue. Later, X. Zhang and C. Song [21] showed that the wheel graph Wn has exactly one positive D-eigenvalue. In this paper, we show that a clique tree has exactly one positive and n − 1 negative D-eigenvalues. n The D-energy is defined in [11] as DE(G) = i=1 |λi (D(G))|. The D-energy, together with a handful of other invariants, has been studied by Consonni and Todeschini [5] for possible use in QSPR modeling. Let Kn1 ,...,nk denote the complete k-partite graph. If ni ≥ 2 for i = 1, · · · , k, Stevanović et al. [17] showed that DE(Kn1 ,...,nk ) = 4(n1 + · · · + nk − k) which is a conjecture proposed by Caporossi et al. [4]. Recently, X. Zhang [19] determined the graphs with the maximum (resp. minimum) D-energy among all complete k-partite graphs with n vertices. For further elaboration, we can recommend a recent book by Li, Shi and Gutman [12]. In this paper, we determine the extremal graphs with maximum and minimum distance energy among all clique trees. 2. The distance matrix of a clique tree is non-singular Let Ja×b be the a × b matrix whose entries are all equal to 1, In be the n × n identity matrix and 1t = (1, . . . , 1). Lemma 2.1. Let G1 be a connected graph and u be a vertex of V (G1 ). Let G be a graph obtained by identifying a vertex of a complete graph Kt and the vertex u of G1 . Then the determinant of the distance matrix of the graph G is fixed, regardless the choice of the vertex of G1 . Proof. Let V (G) = V (G1 − {u}) ∪ V (Kt − {u}) ∪ {u}. Let (0 α) be the row vector of the distance matrix of G1 corresponding to the vertex u and D(G1 − {u}) be the matrix obtained from D(G1 ) by deleting its row and column corresponding to u. Then D(G), the distance matrix of G, can be partitioned as ⎛

A

1

B

⎞

⎜ 1t ⎝ Bt

0

α

⎟ ⎠,

αt

D(G1 − {u})

where A = (J − I)t−1 and B =

α+1t



···

. Hence,

α+1t

A 

t det D(G) = 1

Bt

1 0 αt



α

D(G1 − {u}) B

32

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

−I

= 1t

Bt

1 0 αt

−I − J

= 1t

J



α

D(G1 − {u}) J

1 0 αt



α

.

D(G1 − {u}) J

The last equation implies that det(D(G)) is independent to the choice of the vertex u. Then the result follows. 2 Lemma 2.2. Let G be the graph obtained by identifying k vertices v1 , . . . , vk to a vertex u, where vi ∈ V (Kni ), ni ≥ 2 and n = n1 + n2 + · · · + nk − k + 1. Then det(D(G)) = k k (−1)n−1 i=1 (ni − 1) j=1, j=i nj . Proof. Note that

1t 1t ··· 1t

0



1 (J − I)

2J ··· 2J n1 −1



det D(G) = 1 2J 2J (J − I)n2 −1 · · ·



...

... ... ... ...



1 2J 2J · · · (J − I)nk −1

n2 − 1 ··· nk − 1 n1 − 1

0

1 n1 − 2 2(n2 − 1) · · · 2(nk − 1)



= (−1)n1 +···+nk −k

1 2(n1 − 1) n2 − 2 · · · 2(nk − 1)



... ... ... ... ...



nk − 2 1 2(n1 − 1) 2(n2 − 1) · · ·

n2 − 1 ··· nk − 1

0 n1 − 1

1 n − 2 2(n − 1) · · · 2(n − 1) 1 2 k



= (−1)n+k−1

0 n1 −n2 ··· 0



...

... ... ... ...



0 n1 0 ··· −nk

n1 − 1 n2 − 1 · · · nk − 1



n −n · · · 0 1 2 n+k

= (−1)

... ... ... . . .



n1 0 · · · −nk

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39



n1 − 1 + n1 ki=2 ni −1 n2 − 1 ni

0 −n2 = (−1)n+k

... ...

0 0   k k   ni − 1 n−1 = (−1) ni n1 − 1 + n1 ni i=2 i=2 = (−1)n−1

k  (ni − 1) i=1

k 

nj .

33

· · · nk − 1

··· 0

... . . .

· · · −nk

2

j=1, j=i

By Lemma 2.1 and Lemma 2.2, we present our main theorem in this section. Theorem 2.3. Let G be a clique tree with its cliques Kn1 , . . . , Knk , ni ≥ 2 for i = 1, . . . , k. k k Then det(D(G)) = (−1)n−1 i=1 (ni − 1) j=1, j=i nj . Note that a tree T is a clique tree with k = n − 1 and n1 = · · · = nn−1 = 2. Then we have the following two corollaries. Corollary 2.4. The distance matrix of the line graph of a tree is non-singular. Corollary 2.5. (See [7].) Let T be a tree with order n. Then det(D(T )) = (−1)n−1 × 2n−2 (n − 1). 3. On the inertia of the distance matrix of a clique tree Hermitian matrices have real eigenvalues. The Cauchy interlace theorem states that the eigenvalues of a Hermitian matrix A with order n are interlaced with those of any principal submatrix. Lemma 3.1 (Cauchy interlace theorem). Let A be a Hermitian matrix with order n, and let B be a principal submatrix of A with order m. If λ1 (A) ≥ λ2 (A) ≥ · · · ≥ λn (A) list the eigenvalues of A and μ1 (B) ≥ μ2 (B) ≥ · · · ≥ μm (B) list the eigenvalues of B, then λn−m+i (A) ≤ μi (B) ≤ λi (A) for i = 1, . . . , m. Theorem 3.2. Let G be a clique tree with order n. Then n+ (D(G)) = 1 and n− (D(G)) = n − 1. Proof. Let Kn1 , . . . , Knk be the cliques in G with ni ≥ 2 for i = 1, . . . , k. If k = 1, then the result holds. So in the following, we may assume that k ≥ 2. We use induction on n. Clearly, the result holds for n = 1, 2. So we may assume that the result holds for all clique trees with order less than n ≥ 3. For some i ∈ {1, . . . , k}, we pick a vertex u ∈ V (Kni ) but u ∈ / V (Knj ) for j = i and let G = G − {u}. Then by the hypothesis,

34

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

we have λ1 (D(G )) > 0 and λ2 (D(G )) < 0. Obviously, D(G ) is a principle submatrix of D(G). Then by Lemma 3.1, λ3 (D(G)) ≤ λ2 (D(G )) < 0. By Theorem 2.3, k   det D(G) = (−1)n−1 (ni − 1) i=1

k 

nj .

j=1, j=i

On the other hand, note that n n         n−2 det D(G) = λi D(G) = (−1) λ1 D(G) λ2 D(G) −λi D(G) , i=1

i=3

it follows that λ2 (D(G)) < 0. 2 Subhi and Powers [18] proved that the path Pn has the maximum distance spectral radius among trees with n vertices. It is clear that the complete graph Kn has the minimum distance spectral radius among all graphs with order n. Corollary 3.3. Let L(T ) be the line graph of a tree T with order n. Then L(Pn+1 ) and L(K1,n ) attain the maximum and minimum distance energy among the line graphs of trees, respectively. n Proof. Let T be a tree with order n + 1. Note that i=1 λi (D(L(T ))) = 0. Then by n Theorem 3.2, λ1 (D(L(T ))) = − i=2 λi (D(L(T ))). Then DE(L(T )) = 2λ1 (D(L(T ))). Therefore the result holds. 2 4. On the distance energies of clique trees Let G1 be a connected graph and Pn1 ,...,nk (k ≥ 2) be a clique path. Let G be the graph obtained by identifying a vertex of V (Knk−1 ) in Pn1 ,...,nk−1 , a vertex of V (Knk ) and a vertex v of G1 . Let G be the graph obtained by identifying a vertex of V (Knk ) in Pn1 ,...,nk and a vertex v of G1 . Lemma 4.1. Let G and G be the graphs as shown in Fig. 2. Suppose that V (Knk−1 ) ∩ V (Knk ) ∩ V (G1 ) = vk−1 in the left graph and V (Knk−1 ) ∩ V (Knk ) = vk−1 , V (Knk ) ∩ V (G1 ) = w in the right graph. Then λ1 (D(G)) < λ1 (D(G )). Proof. Let X = (x1 , . . . , xn )t be the Perron vector of G corresponding to λ1 (D(G)). By symmetry, we may suppose that xv = xk for v ∈ V (Knk )\{vk−1 }, xv = xk−1 for v ∈ V (Knk−1 )\{vk−1 , vk−2 } and xvk−1 corresponds to vk−1 . Let u ∈ V (Knk−1 )\{vk−1 }.   Suppose that S1 = v∈V (Kn )∪···∪V (Kn ) xv and S2 = v∈V (G1 )\{vk−1 } xv . Note that 1

k−1

  λ1 D(G) xi = dij xj . vi ∈V

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

35

Fig. 2. A graph transformation from G to G .

Fig. 3. A graph transformation from G to G .

Then we have  λ1 D(G) (xu + xvk−1 − xk ) ≥



d(vk−1 , v)xv + 2(nk − 1)xk − xu

v∈V (G1 )\{vk−1 }

> xk − xu − xvk−1 . It follows that xu + xvk−1 > xk . Therefore      λ1 D G − λ1 D(G) ≥ X t D G − D(G) X = 2S2 (S1 + xvk−1 − xk ) ≥ 2(xu + xvk−1 − xk ) > 0.

2

Lemma 4.2. Let G and G be the graphs as shown in Fig. 3. Then λ1 (D(G)) < λ1 (D(G )). Proof. Let X = (x1 , · · · , xn )t be the Perron vector of D(G) corresponding to λ1 (D(G)). By symmetry, we have xu = x1 for u ∈ V (Kk )\{v}, x2 = xv . Note that |V (G1 )| ≥ 2, then there is a vertex, say w ∈ V (G1 ) such that vw ∈ E(G) and let xw = x3 . Let  S = u∈V (G2 ∪···∪Gs )\{v} xu . Thus we have  λ1 D(G) (x2 + x3 − x1 ) ≥ S + 3x1 − x3 > S + x1 − x2 − x3 .

36

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

It follows that x2 + x3 − x1 > 0. Therefore λ1 (D(G )) − λ1 (D(G)) ≥ x2 + x3 − x1 > 0, that is, λ1 (D(G )) > λ1 (D(G)). 2 Lemma 4.3. We have λ1 (D(Ku,n1 ,...,nk )) ≤ λ1 (D(Ku,2,...,2,n−k )) with equality holding if and only if Ku,n1 ,...,nk ∼ = Ku,2,...,2,n−k . Proof. If k = 1, then the result holds immediately, so we may assume that k ≥ 2. Suppose that X = (x1 , . . . , xn )t is the Perron vector of Ku,2,...,2,n−k corresponding to λ1 (D(Ku,2,...,2,n−k )). By symmetry, we have x1 = xv for v ∈ V (Kn−k ), x2 = xu and x3 = xv for v ∈ / V (G)\({u} ∪ V (Kn−k )). Note that  λ1 D(Ku,2,...,2,n−k ) (2x1 − x3 ) = (2k + 2)x3 + 2x2 − 2x1 > x3 − 2x1 . It follows that 2x1 > x3 . Suppose that n1 = · · · = nt−1 = 2 < nt ≤ · · · ≤ nk . If k = t − 1, then the result holds. Therefore, in the following, we may suppose that k > t − 1. Recall that n − ni − k ≥ 1 for i = t, . . . , k. Then   λ1 D(Ku,n1 ,...,nk ) − λ1 D(Ku,1,...,1,n−k )  ≥ X t D(Ku,n1 ,...,nk ) − D(Ku,1,...,1,n−k ) X ≥ 2x1

k 

  (ni − 1) (n − ni + 1 − k)x1 − x3

i=t

> 2x1 (2x1 − x3 )

k 

(ni − 1) > 0.

2

i=t

Combining Lemma 4.1, Lemma 4.2 and Lemma 4.3, we get the following result. Theorem 4.4. Let G be a clique tree with cliques Kn1 , . . . , Knk . Then λ1 (D(G)) ≥ λ1 (D(Ku,2,...,2,n−k )) with equality holding if and only if G ∼ = Ku,2,...,2,n−k . Combining Theorem 3.2 and Theorem 4.4, we have the following result. Theorem 4.5. Among all clique trees with cliques Kn1 , . . . , Knk , the graph attains the minimum distance energy is Ku,2,...,2,n−k . Lemma 4.6. Let G1 be a connected graph. Let G be the graph obtained by identifying a vertex of G1 and a vertex u of Pn1 ,...,nk for u ∈ V (Knt ) where 2 ≤ t ≤ k − 1. Let G (G ) be the graph obtained by identifying a vertex of G1 and a vertex w of Pn1 ,...,nk for w ∈ V (Kn1 )\{v1 } (w ∈ V (Knk )\{vk−1 }). Then either λ1 (D(G )) > λ1 (D(G)) or λ1 (D(G )) > λ1 (D(G)).

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

37

Proof. Let X = (x1 , . . . , xn )t be the Perron vector of D(G) corresponding to λ1 (D(G)).   Suppose that S1 = v∈V1 ∪···∪Vt−1 xv and S2 = v∈Vt+1 ∪···∪Vk xv . If S1 ≥ S2 , then    λ1 D G − λ1 D(G) ≥ 2



xv (S1 − S2 + xu ) > 0.

v∈V (G1 )

Similarly, if S1 ≤ S2 , then    λ1 D G − λ1 D(G) ≥ 2



xv (S2 − S1 + xu ) > 0.

v∈V (G1 )

Thus, either λ1 (D(G )) > λ1 (D(G)) or λ1 (D(G )) > λ1 (D(G)).

2

Theorem 4.7. Among all clique trees with cliques Kn1 , . . . , Knk , the graph attains the maximum distance spectral radius is Pn1 ,2,...,2,nk . Proof. By Lemma 4.6, the graph attains the maximum distance spectral radius is the clique path. If k = 1, then the result holds. Suppose that G = Pn1 ,n2 ,...,nk for some 2 ≤ t ≤ k such that nt ≥ 3 and V (Kni ) ∩ V (Kni+1 ) = vi for i = 1, . . . , k − 1. Let X = (x1 , . . . , xn )t be the Perron vector of D(G) corresponding to λ1 (D(G)). Suppose   that S1 = v∈V (K  )∪···∪V (K  ) xv and S2 = v∈V (K  )∪···∪V (K  ) xv . Then ein1

nt−1

nt+1

n

k

ther S1 ≥ S2 or S2 ≥ S1 . Without loss of generality, we may assume that S1 ≥ S2 . Then we let G = G − {vvt−1 , vvt | v ∈ V (Knt )\{vt−1 , vt }} + {uv | u ∈ V (Knk ), v ∈ V (Knt )\{vt−1 , vt }}. Then      λ1 D G − λ1 D(G) ≥ X t D G − D(G) X  > (k − 1 − t) nt − 2 (S1 − S2 ) ≥ 0. Thus we have λ1 (D(G )) > λ1 (D(G)). We will do the above graph transformation until n2 = · · · = nk−1 = 2. 2 Combining Theorem 3.2 and Theorem 4.7, we have the following result. Theorem 4.8. Among all clique trees with cliques Kn1 , . . . , Knk , the graph attains the maximum distance energy is Pn1 ,2,...,2,nk . In the following, we will consider the graph with the maximum distance energy among all clique trees. If k = 2, then let G = Pn1 ,n2 and u = V (Kn1 ) ∩ V (Kn2 ). If n1 ≥ n2 + 2, let G = Pn1 −1,n2 +2 . In the following, we will show that λ1 (D(G )) > λ1 (D(G)). Suppose that X = (x1 , . . . , xn )t is the Perron vector of G corresponds to λ1 (D(G)), by symmetry, we have x1 = xv for v ∈ V (Kn1 )\{u} and x2 = xv for v ∈ V (Kn2 )\{u} and x3 = xu . Then  λ1 D(G) x1 = (n1 − 2)x1 + x3 + 2(n2 − 1)x2

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

38

and  λ1 D(G) x2 = 2(n1 − 1)x1 + x3 + (n2 − 2)x2 . Thus we have x1 =

λ1 (D(G))+n2 λ1 (D(G))+n1 x2 .

Therefore

     λ1 D G − λ1 D(G) ≥ 2x1 (n1 − 2)x1 − (n2 − 1)x2 =

(λ1 (D(G)) + n2 )(n1 − 2) − (λ1 (D(G)) + n1 )(n2 − 1) x1 x2 λ1 (D(G)) + n1

(n1 − n2 − 1)λ1 (D(G)) + n1 − 2n2 x1 x2 λ1 (D(G)) + n1  > 0 since λ1 D(G) > n − 1. =

Therefore, by Theorem 3.2, the graph P n+1 , n+1 attains the maximum distance energy 2 2 among all clique trees. If k = 3, then P n2 ,2, n2 attains the maximum distance energy among all clique trees. The result follows by Theorem 3.2 and the following lemma. Lemma 4.9. (See [13, Lemma 3].) λ1 (D(Pn1 ,2,n3 )) ≤ λ1 (D(P n2 ,2, n2 )) with equality holding if and only if |n2 − n3 | ≤ 1. Therefore, we may propose the following conjecture. Conjecture 4.10. Among all clique trees with cliques Kn1 , . . . , Knk , the graph attains the maximum distance energy is P n−k+3 ,2,...,2, n−k+3 . 2

2

Acknowledgement The authors would like to thank the anonymous referee very much for valuable suggestions and corrections which improve the original manuscript. References [1] K. Balasubramanian, Computer generation of distance polynomials of graphs, J. Comput. Chem. 11 (1990) 829–836. [2] R. Bapat, S.J. Kirkland, M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193–209. [3] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan, London, 1976. [4] G. Caporossi, E. Chasset, B. Furtula, Some conjectures and properties on distance energy, Cah. GERAD (2009), G-2009-64. [5] V. Consonni, R. Todeschini, New spectral indices for molecule description, MATCH Commun. Math. Comput. Chem. 60 (2008) 3–14. [6] D.M. Cvetković, M. Doob, I. Gutman, A. Torgašev, Recent Results in the Theory of Graph Spectra, North-Holland, Amsterdam, 1988. [7] R.L. Graham, H.O. Pollack, On the addressing problem for loop switching, Bell Syst. Tech. J. 50 (1971) 2495–2519.

H. Lin et al. / Linear Algebra and its Applications 467 (2015) 29–39

39

[8] I. Gutman, I. Sciriha, On the nullity of line graphs of trees, Discrete Math. 232 (2001) 35–45. [9] A. Ilić, Distance spectral radius of trees with given matching number, Discrete Appl. Math. 158 (2010) 1799–1806. [10] G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Linear Algebra Appl. 430 (2009) 106–113. [11] G. Indulal, I. Gutman, A. Vijaykumar, On the distance energy of a graph, MATCH Commun. Math. Comput. Chem. 60 (2008) 461–472. [12] X.L. Li, Y.T. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [13] H. Lin, L. Feng, The distance spectral radius of graphs with given independence number, submitted for publication. [14] H. Lin, W. Yang, H. Zhang, J. Shu, Distance spectral radius of digraphs with given connectivity, Discrete Math. 312 (2012) 1849–1856. [15] R. Merris, The distance spectrum of a tree, J. Graph Theory 14 (1990) 365–369. [16] H.S. Ramane, D.S. Revankar, I. Gutman, H.B. Walikar, Distance spectra and distance energies of iterated line graphs of regular graphs, Publ. Inst. Math. (Beograd) (N.S.) 85 (2009) 39–46. [17] D. Stevanović, M. Milošević, P. Híc, M. Pokorný, Proof of a conjecture on distance energy of complete multipartite graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 157–162. [18] R. Subhi, D. Powers, The distance spectrum of the path Pn and the first distance eigenvector of connected graphs, Linear Multilinear Algebra 28 (1990) 75–81. [19] X. Zhang, The inertia and energy of distance matrices of complete k-partite graphs, Linear Algebra Appl. 450 (2014) 108–120. [20] X. Zhang, C. Godsil, The inertia of distance matrices of some graphs, Discrete Math. 313 (2013) 1655–1664. [21] X. Zhang, C. Song, The distance matrices of some graphs related to wheel graphs, J. Appl. Math. (2013), Art. ID 707954, 5 pp.