SPECTRAL DISTANCES IN SOME SETS OF GRAPHS

Some of the spectral distance related parameters (cospectrality, spectral eccentricity, and spectral diameter with respect to an arbitrary graph matrix) are determined in one particular set of graphs. According to these results, the spectral distances connected with the adjacency matrix and the corresponding distance related parameters are computed in some sets of trees. Examples are provided of graphs whose spectral distances related to the adjacency matrix, the Laplacian and the signless Laplacian matrix are mutually equal. The conjecture related to the spectral diameter of the set of connected regular graphs with respect to the adjacency matrix is disproved using graph energy.


Introduction
The problems related to the Manhattan spectral distance of graphs were posed by Richard Brualdi at the Aveiro Workshop on Graph Spectra 2006 (see [14]).
Let G i and G j be two non-isomorphic graphs on n vertices whose spectra with respect to some graph matrix M are m 1 (G k ) ≥ m 2 (G k ) ≥ · · · ≥ m n (G k ), k = i, j. The M -spectral distance σ M (G i , G j ) between G i and G j is the Manhattan distance between their M -spectra: In this paper, the spectral distances are considered regarding the adjacency matrix A, the Laplacian L = D − A and the signless Laplacian matrix Q = D + A, where D is the diagonal matrix of vertex degrees. The characteristic polynomial P Gi (x) = det(xI − A) of G i is the characteristic polynomial of its adjacency matrix A, while the eigenvalues of G i connected with the adjacency matrix A are denoted by λ 1 (G i ) ≥ λ 2 (G i ) ≥ · · · ≥ λ n (G i ).
Let I be a family of indices, and G = {G i : i ∈ I} a set of non-isomorphic graphs of order n. The M -cospectrality of G i ∈ G is cs M G (G i ) = min{σ M (G i , G j ) : G j ∈ G, i = j}, while the M -cospectrality measure of the set G is cs M (G) = Conjecture 1.1. Let R 1 and R 2 be the graphs having the maximum A-spectral distance among the connected regular graphs of order n. Then, one of them is the complete graph on n vertices.
In this paper, the parameters of spectral distances, namely the M -cospectrality, the M -spectral eccentricity and the M -spectral diameter, will be computed in some particular sets of graphs, and Conjecture 1.1 will be disproved.
The notation common for spectral graph theory is used in the paper. In that way, P n is the path on n vertices, C n is the cycle of order n, K n is the complete graph on n vertices, while K 1,n is the star of order n + 1. For the complete multipartite graph on p parts and k vertices in each of them, i.e. for the complete p-partite graph, the expression K k, k, . . . , k p or K p×k is utilized.
The complement of G is denoted by G. The graph G i ∪ G j means the disjoint union of the graphs G i and G j , while the disjoint union of k copies of the same graph G i is labeled by k G i . The join G i ∇G j of disjoint graphs G i and G j is the graph obtained from G i ∪ G j by joining each vertex of G i to each vertex of G j . The following result will be used in some proofs in Section 3. Theorem 1.2 ([5, Theorem 2.1.8]). If G i is a r i -regular graph with n i vertices, and G j is a r j -regular graph with n j vertices, then the characteristic polynomial of the join G i ∇G j is given by SPECTRAL DISTANCES IN SOME SETS OF GRAPHS 3 The energy E(G i ) of the graph G i with respect to the adjacency matrix is defined as E(G i ) = n s=1 |λ s (G i )|. For the remaining notation and terminology, the reader is referred to [4] or [5].
The paper is organized as follows: in Section 2, some of the parameters of the M -spectral distances (the M -cospectrality, the M -spectral eccentricity, and the M -spectral diameter) are determined in one particular set of graphs. According to these results, the A-spectral distance related parameters in some sets of trees are computed. In Section 3, some examples of graphs whose A-, L-and Q-spectral distances are mutually equal are provided, while in Section 4, Conjecture 1.1 is disproved using graph energy.

Spectral distances in some sets of trees
The triangular inequality implies the following result.
. . , G f (n) } be the set of non-isomorphic graphs on n vertices, and let m 1 (G k ) ≥ m 2 (G k ) ≥ · · · ≥ m n (G k ), k = 1, 2, . . . , f (n), be the M -spectra of these graphs with respect to some graph matrix M . Then It is obvious that the equality in (1) in Proposition 2.1 is attained if the graphs G i , G j and G k are mutually cospectral, or if G k is cospectral with one of the graphs G i or G j . One more case in which the equality in (1) in Proposition 2.1 is attained is given by the next proposition.
, for each s = 1, 2, . . . , n. Then The spectral distance related parameters on the set of graphs of equal order, regarding an arbitrary graph matrix, whose spectral distances satisfy the equality stated in Proposition 2.2, have been computed by the following proposition.
. . , G f (n) } be the set of non-isomorphic graphs of order n and let M be an arbitrary graph matrix. If the M -spectral distances of graphs from the set G satisfy (2.1), then the parameters of M -spectral distances on the set G are:

and in particular
Now, the spectral eccentricities and the spectral diameter can be determined.
For fixed 3 ≤ j ≤ f (n) and for each 1 ≤ i ≤ j − 2, the following inequality holds: wherefrom one obtains and the cospectralities can be calculated. Proof. Let G 1 , G 2 , . . . , G n 2 be the sequence of graphs given by G 1 = K 1,n and G i = K 1,n−i+1 ·K 1,i−1 , for i = 2, 3, . . . , n 2 . This means that T = {G 2 , . . . , G n 2 }. Using Theorem 1.3, one finds the characteristic polynomial of G i with respect to the adjacency matrix: Then, the spectrum of G i is ± λ 1 (G i ), ± λ 2 (G i ), and [0] n−3 , where λ 1 (G i ) = be checked that λ 1 (G i ) ≥ λ 1 (G j ) and λ 2 (G i ) ≤ λ 2 (G j ), for 1 ≤ i < j ≤ n 2 , so the A-spectral distance between G i and G j , 1 ≤ i < j ≤ n 2 , is Remark 2.6. It is obvious that the parameters of the A-spectral distances of graphs from the set T are also given by Proposition 2.4.
The graphs from the set T • for n = 7 are presented in Figure 1. Their adjacency eigenvalues (rounded to two decimal places) are: Spect(G  Let T * = {G 0 , G 1 , . . . , G n 2 } be the set of graphs of order n + 1, where n > 4 is even, Here, for fixed i, by the graph operation of coalescence, a vertex of the j-th graph K 2 is identified with a pendant vertex of the graph ((( , which is adjacent to the vertex of the maximum degree (i.e. n − i) in this graph, for each 1 ≤ j ≤ i. It can be noticed that the graphs from the set T * are such trees that G i has i vertices of degree 2, one vertex of degree n − i and n − i pendant vertices. The graphs from T * for n = 8 are presented in Figure 2.
In order to determine the characteristic polynomials of the graphs from the set T * regarding the adjacency matrix, one can use G Lemma 2.7. The characteristic polynomials of the graphs from the set T * , with respect to the adjacency matrix, are given by the recurrent formula i−1 which is adjacent to the vertex of the maximum degree in this graph, according to Theorem 1.3, one can get: The explicit formula for the characteristic polynomial of G Proof. The following equality holds: where it is supposed that 0 0 = 1. Further, one finds: Proof. The characteristic polynomial of G (n+1) 1 regarding the adjacency matrix is so the spectrum of G  with respect to the adjacency matrix is Therefore, the spectrum of G i.e. that the characteristic polynomial of the graph G The characteristic polynomial of the graph G (n+1) l+1 , according to (2.2), satisfies the relation wherefrom, by using (2.4), one can get

Considering (2.3), the previous relation becomes:
which means that the statement is proved by using the principle of mathematical induction.

SPECTRAL DISTANCES IN SOME SETS OF GRAPHS 9
The roots of the polynomial Q (n+1) i (x) are By direct computation, it can be checked that x 1 > 1 and x 2 < 1, i.e. x 3 < −1 and In [13], the authors consider the graphs G k,l (1 ≤ k ≤ l) and G k−1,l+1 , defined as follows: let G be a connected graph with at least two vertices and let v be an arbitrary vertex. G k,l is the graph obtained by coalescing G with two new paths, by identifying an end vertex of P k+1 and an end vertex of Q l+1 , both with the In this regard, the authors state the following lemma from [12]. . Let G be a connected graph on n (n ≥ 2) vertices and v( v 0 ) be a vertex of G, and let G k,l (l ≥ k ≥ 1) be the graph defined as above. Then, It can be noticed that for each 0 ≤ i ≤ n 2 − 1, the graph G (n+1) i from the set T * is, in fact, the graph G k,l for k in the manner described above. So, according to Lemma 2.11, it follows that By direct computation, one can verify that the A-spectral distances between graphs from the set T * satisfy (2.1). In that way, the following theorem has been proved. Therefore, the A-spectral distances between these graphs are approximately equal to: The following lemma and conjecture appear in [9].
The next statement supports Conjecture 2.14: Proposition 2.15. Let T n and T * n be two trees of order n such that The following equality holds: so one can obtain: since λ 1 (K 1,n−1 ) ≥ λ 1 (T n ) for every tree T n of order n. The previous inequality together with Lemma 2.13 means that 3. Examples of graphs whose spectral distances with respect to the adjacency, the Laplacian and the signless Laplacian matrix are mutually equal In [10], an example of a graph whose A-, L-and Q-spectral distances are mutually equal is presented. Precisely, the following proposition has been proved.
In this section, some new families of graphs whose A-spectral distances are equal to the corresponding L-and Q-spectral distances will be considered. First, let us recall that the graph G 2 is an edges-deleted subgraph of the graph G 1 if it is obtained by deleting some edges of G 1 and that the following result holds.  by deleting k edges from G 1 , then σ L

Proposition 3.3.
Let G 1 and G 2 be two graphs of order n such that G 2 is an edges-deleted subgraph of G 1 , and ε(G 1 ) − ε(G 2 ) = k, where ε(G i ), i = 1, 2, is the number of edges of the graph G i . Let the A-spectra of these graphs be λ 1 (G i ) ≥ λ 2 (G i ) ≥ · · · ≥ λ n (G i ), i = 1, 2, and let I ⊆ {1, 2, . . . , n} be the set of indices i Proof. The spectral distance between G 1 and G 2 regarding the adjacency matrix is equal to since the sum of the adjacency eigenvalues of a graph is equal to zero. The remaining two equalities follow from Proposition 3.2.
Let K = {G 0 , G 1 , . . . , G q } be the set of graphs of order n+1 such that G 0 = K 1,n and G i = K 1,(n−im) · H i , i = 1, 2, . . . , q. Here, H i = K 1 ∇iK m ; by the graph operation ·, the vertex of maximum degree in the graph K 1,(n−im) is identified with the vertex of maximum degree in the graph H i ; n ≡ 0 (mod m), i.e. n = m q for q ∈ Z + ; n > m and m > 1. Graph G 2 from K for n = 16 and m = 4 is presented in Figure 3.
To prove Proposition 3.6, which is related to the adjacency spectral distances of graphs from the set K, one needs the following results. . Let G be a graph with n vertices and eigenvalues λ 1 ≥ λ 2 ≥ · · · ≥ λ n , and let H be an induced subgraph of G with m vertices. If the eigenvalues of H are µ 1 ≥ µ 2 ≥ · · · ≥ µ m , then λ n−m+i ≤ µ i ≤ λ i , i = 1, 2, . . . , m.  (1) Let G i , G j and G k be three arbitrary graphs from the set K, such that 1 ≤ i < j < k ≤ q. Then the A-spectral distances of these graphs satisfy (2.1). (2) The A-, L-and Q-spectral distances of graphs from K are mutually equal for m = 2.
Proof. By applying Theorem 1.2, the characteristic polynomial of the graph H i = K 1 ∇iK m , with respect to the adjacency matrix, can be obtained: According to Theorem 1.3, the characteristic polynomial of the graph G i , regarding the adjacency matrix, is . Let x 1 , x 2 and x 3 be the roots of the polynomial Q i (x). According to Corollary 3.4, as K m is an induced subgraph of Graph H i is an induced subgraph of G i , for each 1 ≤ i < q, and from (3.1) it follows that its A-spectrum is Applying Corollary 3.4 results in: the A-spectrum of K 1 ∇q K m−1 consists of the following eigenvalues: The functions Q i (x) and Q j (x) differ only in the positive coefficient with x 0 and they have no common points, which means that one of them is the expansion (shrinking) of the other in a certain interval. The function Q i (x), 1 ≤ i ≤ q, is positive in the interval (x 2 , x 3 ), it is monotonically increasing on −∞, 1 3 m − 1 − (m − 1) 2 + 3n , and it attains the maximum for x = 1 3 m − 1 − (m − 1) 2 + 3n . Hence, and also For G i , G j ∈ K, where 1 ≤ i < j ≤ q, the following holds: and The relations (3.4) and (3.5) and Proposition 3.3 imply that the A-, L-and Q-spectral distances of graphs from the set K are mutually equal for m = 2 (see Figure 4).
Otherwise, the A-spectral distance of graphs G i and G j , for 1 ≤ i < j ≤ q, is equal to wherefrom it can be checked that (2.1) is satisfied. There are examples of graphs such that the A-spectral distance between them and their edge-deleted subgraphs is equal to a small constant. For example, in [9] it was proved that σ A (C n , P n ) = σ A (C n , C n −e) = 2, while in [1] it was proved that σ A (K n , K n − e) = 2, for n ≥ 2. Here, G − e stands for an edge-deleted subgraph of the graph G. A similar situation, which holds in the case of the complete multipartite graph and its edge-deleted subgraph, is described by Proposition 3.9. However, two statements relevant for proving this proposition will be presented first. The reader who needs to be reminded of the adjacency spectrum of the complete multipartite graph is referred to [4, p. 73].
Moreover, for t − 1 of the negative eigenvalues λ i (i = 2, . . . , t), the strict inequalities hold, and together with λ 1 they are the roots of the equation Proposition 3.9. Let G = K p×k be the complete p-partite graph of order n = p k, and let λ 1 (G − e) ≥ λ 2 (G − e) ≥ · · · ≥ λ n (G − e) be the A-spectrum of the graph G − e such that λ 2 (G − e) + λ n−p+2 (G − e) = 1 − k. Then the following holds: Proof. The graph G * = K The remaining two equalities follow from Proposition 3.2.
Let G be a graph with vertex set V (G). The partition V (G) = V 1∪ V 2∪ · · ·∪ V k , where∪ stands for the disjoint union, is an equitable partition if every vertex in V i has the same number of neighbours in V j , for all i, j ∈ {1, 2, . . . , k} (for more details see e.g. [5, p. 83]).
The following statements, used in Remark 3.12 below, were proved in [5].
The i-th row and column of Q correspond to C i , i = 1, 2, 3, 4. The characteristic polynomial of the matrix Q is q(x) = x 4 − (n 1 n 2 − 1) x 2 + (n 1 − 1) (n 2 − 1), while its eigenvalues are and According to Theorem 3.11, these eigenvalues are also the A-eigenvalues of the graph K n1,n2 − e. Since K n1,n2−1 is an induced subgraph of K n1,n2 − e, from Corollary 3.4 it follows that in the A-spectrum of K n1,n2 − e there are n 1 + n 2 − 4 eigenvalues which are equal to 0. Since it follows that σ A (K n1,n2 , K n1,n2 − e) = 2 only if n 1 ≥ 1 and n 1 = n 2 .
Proposition 3.13. For n > 2 and m ≥ 2, the following holds: Proof. By applying Theorem 1.2, one can easily obtain that the A-spectrum of K n ∇K m,m consists of the following eigenvalues: The adjacency matrix A((K n − e)∇K m,m ) of the graph (K n − e)∇K m,m has the blocking as follows: while the characteristic polynomial of the matrix Q is The roots y 1 , y 2 , y 3 of the polynomial q(x) are eigenvalues of (K n − e)∇K m,m . The other eigenvalues of (K n − e)∇K m,m remain the same if J is subtracted from the blocks of A((K n − e)∇K m,m ) which are equal to J, and from the block which is equal to J − I (for more details related to this subject, see [3]). Thus, one obtains: In what follows, the order of the eigenvalues y 1 , y 2 , y 3 in the spectrum of the graph (K n − e)∇K m,m will be determined. One of them, say y 1 , is obviously the index of the graph. Since K n−1 ∇K m,m is an induced subgraph of (K n − e)∇K m,m , according to Corollary 3.4 the remaining two eigenvalues, y 2 and y 3 , may be in one of the following intervals (each of y 2 and y 3 in exactly one interval): [0, are the corresponding eigenvalues of the graph K n−1 ∇K m,m . Since the following relations hold: 4. About the conjecture related to the adjacency spectral distances of connected regular graphs In [9], Z. Stanić posed the following six conjectures related to the A-spectral distances of graphs: Conjecture 4.1. The A-spectral distance between any two graphs of order n does not exceed E max n = max{E(G) : G is a graph of order n}, i.e.
where G n is the set of all n-vertex graphs.

Conjecture 4.2.
Let R 1 and R 2 be the graphs having the maximum A-spectral distance among the connected regular graphs of order n. Then one of them is K n , i.e.
sdiam A (R n ) = secc A Rn (K n ), where R n is the set of all connected regular graphs of order n.   where Z n and W n are, respectively, the snake graph and the double snake graph of order n.
Conjecture 4.6. For any > 0, there are graphs G 1 and G 2 having no common eigenvalues such that σ A (G 1 , G 2 ) < .
Conjecture 4.1 was disproved in [8], and now Conjecture 4.2 will be considered. Let R n be the set of connected regular graphs of order n, and let G ∈ R n be a r-regular graph whose adjacency spectrum is r = λ 1 (G) ≥ λ 2 (G) ≥ · · · ≥ λ n (G).
One can easily verify that where G is the complement of G, and G is (n − 1 − r)-regular. Also, G = G.
Since the energy E(G) of a graph G on n vertices is not greater than n 2 (1 + √ n) (see [11]), one can conclude that which means, according to the statement given by Conjecture 4.2, that The generalized quadrangle G = GQ (3,9) (for more details about this kind of graphs, see [7]) is the strongly regular graph with parameters (112, 30, 2, 10), whose A-spectrum is 30, [2]  and since both graphs, G = GQ (3,9) and its complement G, are connected, Conjecture 4.2 is disproved.