Some properties of eccentrical graphs

In this paper, the concept of the eccentrical graph of a graph is introduced. Let G be a connected graph with the vertex set V ( G ) . The eccentrical graph of G is the graph (cid:15) ( G ) with the vertex set V ( (cid:15) ( G )) = V ( G ) and two vertices v i , v j ∈ V ( (cid:15) ( G )) are adjacent in (cid:15) ( G ) if and only if the distance between them is min { e ( v i ) , e ( v j ) } , where e ( v i ) is the eccentricity of v i . A suﬃcient condition for the eccentrical graph of a connected graph to be connected is given. It is proved that the eccentrical graph of every tree is connected and its diameter does not exceed 3 . The extremum values of the greatest eigenvalue of eccentrical graphs of trees and connected graphs of ﬁxed order are also studied. Furthermore, spectra of eccentrical graphs of various classes of graphs are computed.


Introduction
Let G = (V (G), E(G)) be a finite simple graph with vertex set V (G) = {v 1 , v 2 , . . ., v n } and edge set E(G) = {e 1 , . . ., e m }.The notion v i ∼ v j is used to indicate that vertices v i and v j are adjacent, and the edge between them is denoted by v i v j or e ij .The degree of the vertex v i of G is denoted by deg(v i |G).The adjacency matrix of G is an n × n matrix, denoted by A(G), whose rows and columns are indexed by the vertex set of G and its entries are defined by The distance between the vertices v i , v j ∈ V (G), denoted by d(v i , v j ), is defined as the smallest value among the lengths (i.e., the number of edges) of the paths between the vertices v i and v j .The distance matrix of a connected graph G, denoted by D(G), or simply by D, is the n × n matrix whose (i, j) th -entry is equal to d(v i , v j ), where i = 1, 2, . . ., n, and j = 1, 2, . . ., n.For other terminologies and notations not defined here, the readers are referred to [2].The adjacency matrix and the distance matrix of a graph are well-studied matrices in the field of spectral graph theory.Details about the study of these matrices and other matrices associated with graphs can be found in [1,3,4].
The eccentricity e(v i ) of the vertex v i is defined as e(v i ) = max{d(v i , v j ) : 1.1 for two examples).Thus, the (i, j) th -entry of the adjacency matrix of the eccentrical graph (G), denoted by A (G), is given as follows: Let 1 , 2 , . . .n denote the eigenvalues of the matrix A (G). Since, the matrixA (G) is symmetric, all the -eigenvalues of G are real.If 1 , 2 , . . ., k are the all distinct -eigenvalues of G satisfying 1 > 2 > . . .> k , then the -spectrum of G is denoted by where m i is the algebraic multiplicity of the eigenvalue i for 1 ≤ i ≤ k.The eigenvalue 1 is known as the spectral radius of the eccentrical graph (G).
The eccentrical graph of G 1 The eccentrical graph of C 6 C 6 Figure 1.1: Two graphs and their eccentrical graphs.
The complement G of a graph G is the graph whose vertex set is the same as that of G and two vertices are adjacent in G if and only if they are not adjacent in G.
is the graph obtained from G 1 ∪ G 2 by making every vertex of G 1 adjacent to all vertices of G 2 .The join operation of two graphs is also known as the complete product of two graphs.The corona of G 1 and G 2 , denoted by G 1 • G 2 , is defined as the graph obtained by taking one copy of G 1 and n 1 copies of G 2 , and then making the i-th vertex of G 1 adjacent to every vertex in the i-th copy of G 2 .
This article is organized as follows.In Section 2, a sufficient condition for the eccentrical graph of a connected graph to be connected is given.It is also proved in Section 2 that the eccentrical graph of every tree is connected and its diameter does not exceed 3.In Section 3, the extremum values of the spectral radius of eccentrical graphs of trees and connected graphs of fixed order are investigated.Spectra of eccentrical graphs of various classes of graphs are also computed in Section 3.

The eccentrical graph of a tree
For a symmetric matrix M of order n, its matrix graph G M is the graph whose vertices are 1, 2, . . ., n, and two distinct vertices i, j are adjacent if and only if [7].By Figure 1.1, the eccentrical graph of a connected graph may be disconnected.Theorem 2.1.Let G be a connected graph of order n.Let P uv be a path of longest length in G with end vertices u and v.If for every vertex w ∈ V (G), it holds that e(w) = max{d(w, u), d(w, v)}, then the eccentrical graph of G is connected.
Proof.For any s, t ∈ V (G), we have e(s) = max{d(s, u), d(s, v)} and e(t) = max{d(t, u), d(t, v)}.By the definition of the eccentrical graph, at least one of the two edges su(tu) and sv(tv) belongs to E( (G)), and also uv ∈ E( (G)).Therefore, the eccentrical graph of G is connected.Theorem 2.2.If T is a tree of order n, then the eccentrical graph (T ) of T is connected and the diameter of (T ) does not exceed 3.
Proof.Let P uv be a path of the longest length in T with end vertices u and v. Consider a vertex w ∈ V (T ) such that e(w) = d(w, u) and e(w) = d(w, v).Then there is a vertex s ∈ V (T ) such that d(w, s) > d(w, u) and d(w, s) > d(w, v).
Case 1: w ∈ P uv .In this case, either the path P vs or the path P us has length larger than the length of the path P uv in T , which is not possible.

Subcase 2.1:
which is again a contradiction.Therefore, by the above discussion and Theorem 2.1, the eccentrical graph (T ) is connected.Also, by the definition of the eccentrical graph, the diameter of (T ) does not exceed 3.

The spectral radius of the -spectrum of graphs
The following theorem is known as the interlacing theorem.Lemma 3.1 (see [7]).Suppose that A ∈ R n×n is symmetric.Let B ∈ R m×m , with m < n, be a principal submatrix of A (submatrix whose rows and columns are indexed by the same index set {i 1 , . . ., i m }, for some m).Suppose that λ 1 , . . ., λ n are the eigenvalues of A such that λ 1 ≤ . . .≤ λ n and β 1 , . . ., β m are the eigenvalues of B satisfying By Lemma 3.1, we have the next result.Lemma 3.2 (see [4]).Let G be a graph with n vertices and m edges, where m > 1.Let λ 1 , λ 2 , . . ., λ n denote the eigenvalues of the adjacency matrix Let λ(G) be the radius adjacency spectrum of a graph G.

Proof. (a)
. By the definition of the eccentrical graph, we have that and the eigenvalues of the matrix A (C 2k+1 ) are i = 2 cos 2πi 2k+1 where i = 1, 2, . . ., 2k + 1. (b).By the definition of the eccentrical graph, we have that (K n−k ∨ kK 1 ) ∼ = K n and hence Let BS(n 1 , n 2 ) be the double star with n = n 1 + n 2 + 2 vertices.Denote by BBS(n) the balance double star with n vertices.Theorem 3.2.Let T be a tree with n vertices, where n ≥ 7. Let 1 (T ), 2 (T ), . . ., n (T ) denote the eigenvalues of the matrix A (T ) such that 1 (T ) ≥ 2 (T ) ≥ . . .≥ n (T ).Then λ(BBS(n)) ≤ 1 (T ) ≤ n − 1, with the left equality if T ∼ = T * (see Figure 3.1) and with the right equality if Proof.By Lemma 3.1 and Lemma 3.2, we have By Theorem 2.2, the diameter of (T ) is at most 3.If the diameter of (T ) is at most 2, we have If the diameter of (T ) is 3, then by the definition of the eccentrical graph and Theorem 2.2, there is a double star BS(n 1 , n 2 ) 3.1), by the definition of the eccentrical graph we have (T * ) ∼ = BBS(n).
Proof.Let d be the diameter of the graph G. Let P (v 1 , v d ) be a path of the longest length in G.
By Theorem 3.1 and Lemma 3.2, we have .
By the definition of the eccentrical graph, The vertices v 1 and v d are adjacent in (G).By Lemma 3.1, we have that Conjecture 3.1.Let G be a connected graph with n vertices.Let 1 (G), 2 (G), . . ., n (G) denote the eigenvalues of the matrix

The -spectrum of some classes of graphs
Let A be an n × n matrix partitioned as where A 11 and A 22 are square matrices.If A 11 is nonsingular, then the Schur complement of A 11 in A is defined as Lemma 3.3 (see [5,9]).Let B = B 0 B 1 B 1 B 0 be a symmetric 2 × 2 block matrix such that B 0 and B 1 are square matrices of the same order.Then, the spectrum of B is the union of the spectra of B 0 + B 1 and B 0 − B 1 .
Lemma 3.4 (see [9]).Let B be a square matrix of order n.If each column sum of B is equal to some eigenvalue (say α) of B, then The following result is about the spectrum of a special kind of block matrices.
Lemma 3.5.Let A be an (n + 1) × (n + 1) matrix of the form Proof.The characteristic polynomial of A is given by By Schur complement formula and Lemma 3.4, we have which gives the required result.
Next, we compute -eigenvalues of some classes of graphs.First, we compute the -spectrum of the corona of any connected graph G with the complete graph on n vertices.Theorem 3.4.If K n is the complete graph on n vertices and G is any connected graph on m vertices, then where λ 1 and λ 2 are the roots of x 2 − mx − m = 0.
Proof.Let K n be the complete graph on n vertices and let G be any connected graph on m vertices.Then, the graph K n • G consists of n vertices of the complete graph K n which are labeled using the index set {1, 2, . . ., n}, and n disjoint copies G 1 , G 2 , . . ., G n of G. Choose an arbitrary ordering g 1 , g 2 , . . ., g m of the vertices of G, and label the vertices of G i corresponding to g k by the indices i + nk (see [8]).Under this labeling, the adjacency matrix of the eccentrical graph of where By Lemma 3.5, we have 1 .
Now, by Lemma 3.6, the spectrum of and hence Next, we consider the -spectrum of the complete product of two graphs.Proof.By the definition of the eccentrical graph, we have that Therefore, the -spectrum of G 1 ∨ G 2 is the union of the spectra of A(G 1 ) and A(G 2 ).. Lemma 3.7 (see [4]).Let G i be a connected r i -regular graph with n i vertices, where i = 1, 2. The characteristic polynomial of the complete product of G 1 and G 2 is (b).If r = n − 1, then by the definition of the eccentrical graph, we have where By Lemma 3.4, we have By Schur complement formula, we have

Theorem 3 . 5 .
Let G 1 and G 2 be any two non-complete connected graphs.If the eigenvalues of the adjacency matrices of G 1 and G 2 are known, then the -spectrum of G 1 ∨ G 2 is the union of the spectra of A(G 1 ) and A(G 2 ).
Similarly, if A 22 is nonsingular, then the Schur complement of A 22 in A is A 11 − A 12 A −1 22 A 21 , and we have det A = (det A 22 ) det(A 11 − A 12 A −1 22 A 21 ).