On the diameter and incidence energy of iterated total graphs

The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is the union of the sets of vertices and edges of $G$, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in $G$. Let $\mathcal{T}^1(G)=\mathcal{T}(G)$, the total graph of $G$. For $k\geq2$, the $k\text{-}th$ iterated total graph of $G$, $\mathcal{T}^k(G)$, is defined recursively as $\mathcal{T}^k(G)=\mathcal{T}(\mathcal{T}^{k-1}(G)).$ If $G$ is a connected graph its diameter is the maximum distance between any pair of vertices in $G$. The incidence energy $IE(G)$ of $G$ is the sum of the singular values of the incidence matrix of $G$. In this paper for a given integer $k$ we establish a necessary and sufficient condition under which $diam(\mathcal{T}^{r+1}(G))>k-r,$ $r\geq0$. In addition, bounds for the incidence energy of the iterated graph $\mathcal{T}^{r+1}(G)$ are obtained, provided $G$ to be a regular graph. Finally, new families of non-isomorphic cospectral graphs are exhibited.


Introduction
Let G be a simple connected graph on n vertices. Let v 1 , v 2 , . . . ,v n be the vertices of G. An edge e with end vertices u and v will be denoted by (uv). Sometimes, after a labeling of the vertices of G, a vertex v i is simply referred by its label i and an edge v i v j is simply referred as (ij). The incident vertices to the edge (ij) are i and j. The distance between two vertices v i and v j in G is equal to the length of the shortest path in G joining v i and v j , denoted by d G (v i , v j ). The diameter of G, denoted by diam(G), is the maximum distance between any pair of vertices in G.
The above distance provides the simplest and most natural metric in graph theory, and its study has recently had increasing interest in discrete mathematics research. As usual, K n , P n , C n , and S n denote, respectively, the complete graph, the path, the cycle, and the star of n vertices.
The line graph of G, denoted by L(G), is the graph whose vertex set are the edges in G, where two vertices are adjacent if the corresponding edges in G have a common vertex. The k-th iterated line graph of G is defined recursively as L k (G) = L(L k−1 (G)), k ≥ 2, where L(G) = L 1 (G) and G = L 0 (G). Metric properties of the line graph have recently been extensively studied in the mathematical literature [3,6,13,15,21], and it found remarkable applications in chemistry [11,14,16,18].
The total graph of G, denoted by T (G), is the graph whose vertex set corresponds union of set of vertices and edges of G, where, two vertices are adjacent if their corresponding elements are either adjacent or incident in G. The k-th iterated total graph of G is defined recursively as T k (G) = T (T k−1 (G)), k ≥ 2, where T (G) = T 1 (G) and G = T 0 (G).
The incidence matrix, I(G), is the n × m matrix whose (i, j)-entry is 1 if v i is incident to e j and 0 otherwise. It is known that and where Q(G) is the signless Laplacian matrix of G and A(L(G)) is the adjacency matrix of the line graph of G.
In [12,19], the authors have introduced the concept of the incidence energy IE(G) of G as the sum of the singular values of the incidence matrix of G. It is well known that the singular values of a matrix M are the nonzero square roots of M M T or M T M as these matrices have the same nonzero eigenvalues. From these facts and (1) it follows that where q 1 , q 2 , . . . ,q n are the signless Laplacian eigenvalues of G.
In [12], Gutman, Kiani, Mirzakhah, and Zhou, proved that Theorem 1.1 [12] Let G be a regular graph on n vertices of degree r. Then, where n k and r k are the order and degree of L k+1 (G), respectively. Equality holds if and only if This paper is organized as follows. In Section 2, we establish conditions on a graph G, in order to the diameter of T (G) does not exceed k, k ≥ 2. Also conditions in order to the diameter of T (G) to be greater than or equal to k, k ≥ 3, are established. Moreover, we establish a necessary condition so that the diameter of T r+1 (G) does not be greater than or equal to k − r, k ≥ 2, r ≥ 0. In Section 3, we derive upper and lower bounds on the incidence energy for the iterated total graphs of regular graphs. Aditionally, we construct some new families of nonisomorphic cospectral graphs.

Diameter of total graphs
In this section we establish structural conditions for a graph G, so that the diameter of T (G) does not exceed k and it be no less than k, k ≥ 3.

Main results
Before proceeding we need the following definitions.
Definition 2.1 A path P with end vertices u and v in a connected graph G is called a diameter path of G if d G (u, v) = diam(G) and P is a path with diam(G) + 1 vertices.
has a diameter path of G as a subgraph.

Definition 2.3 The Lollipop
Lol n,g is the graph obtained from a cycle with g vertices by identifying one of its vertices with a vertex of a path of length n − g. Note that this graph has n vertices and diameter n − g + 1 + g 2 . In [5] was conjectured that Lol 3,n−3 is the nonbipartite graph on n vertices with minimum smallest signless Laplacian eigenvalue q n .  or diam(T (G)) = diam(L(G)) or Lol l+diam(G)+1,2l+1 is a diameter subgraph of G for some Proof. The result can be easily verified for graphs of order n ≤ 3. In consequence, we assume, n > 3. If diam(G) < diam(T (G)), diam(L(G)) < diam(T (G)) and Lol l+diam(G)+1,2l+1 is not a diameter subgraph of G for all 1 ≤ l ≤ diam(G) where diam(T (G)) = diam(G) + 1. We claim that d T (G) (u, v) < diam(T (G)) for any pair of vertices u, v in T (G). Let v a , v b be two vertices in . Then there must exists diam(T (G)) − 1 vertices in where v a is adjacent to v 1 .
1. If v a , v b are vertices of G then we can assume that v i is a vertex of G, i = 1, 2, . . . ,diam(T (G))− 1. Then, diam(G) ≥ diam(T (G)), which is impossible.
2. If v a , v b are vertices of the line graph of G, that is to say if v a , v b are edges of G, then we can assume that v i is an edge of G, i = 1, 2, . . . ,diam(T (G)) − 1. Thus, diam(L(G)) ≥ diam(T (G)), which is impossible.
3. If v a is a vertex of G and v b is an edge of G, say v b = (b 1 b 2 ), we can assume without loss of generality v i is a vertex of G, i = 1, 2, . . . ,diam(T (G)) − 1, or v i is an edge of G, i = 1, 2, . . . ,diam(T (G)) − 1. Suppose v i is a vertex of G, i = 1, 2, . . . ,diam(T (G)) − 1.
Before proceeding we need establish the following facts about the line graphs.
Let F k 2 and F k 3 be the graphs obtained from F k+1 1 by adding the edge (v 1 v 3 ), and the edges • F k 1 is a diameter path of G, and Proof. The result can be easily verified for graphs of order n ≤ 3. In consequence, we assume, which is a contradiction. Conversely, suppose that k ≥ 2 and diam(T (G)) > k. By Lemma 2.1, we have the following cases is an induced subgraph of G, a contradiction.
An equivalent result to Theorem 2.1 is given below.
For a connected graph G, diam(T (G)) > k if and only if some of the following conditions is verified Proof. Let G be a connected graph such that diam(L(G)) = 2 then there exists 4 vertices, say u 1 , u 2 , u 3 , u 4 , such that u i is adjacent to u i+1 , for i = 1, 2, 3.
(i) If u 1 is not adjacent to u 3 , u 2 is not adjacent to u 4 and u 1 is not adjacent to u 4 then P 4 is an induced subgraph of G. Moreover, since diam(G) = 2 then P 3 : u 1 vu 4 is a diameter path of G for some v. Hence, C 5 is a diameter subgraph of G.
(ii) If u 1 is adjacent to u 3 (or u 2 is adjacent to u 4 ) and u 1 is not adjacent to u 4 then Lol 4,3 is an induced subgraph of G.
(iii) If u 1 is adjacent to u 3 , u 2 is adjacent to u 4 and u 1 is not adjacent to u 4 then K 4 − e is an induced subgraph of G.
(iv) If u 1 is adjacent to u 4 , u 1 is not adjacent to u 3 and u 2 is not adjacent to u 4 then C 4 is an induced subgraph of G.
Theorem 2.4 Let G be a connected graph that such all the following conditions fail to hold In consequence, G ∼ = K 3 or G ∼ = S n . Since diam(G) = 2, we concluded that G ∼ = S n .

Results for iterated total graphs
Theorem 2.5 Let r ≥ 1 and k ≥ 4r + 3. Let G be a connected graph such that diam(T r+1 (G)) > k − r.
Then F k−4r−1 1 is an induced subgraph of G.
Proof. Suppose diam(T r+1 (G)) > k − r. By Theorem 2.2, Moreover, is an induced is a diameter path of L(T r−1 (G)). Since L(F k−r+1 is an induced subgraph of T r−1 (G).
is a diameter path of is an induced subgraph of T 2 (G). Following this process we concluded is an induced subgraph of T r−1 (G). Then, the graph with vertices is an induced subgraph of T r (G) isomorphic to F k−r

Results for iterated line graphs
k ≥ 3. Let F k 4 be the graph obtained from P k−1 by joining two new vertices to the vertex v 1 and another two new vertices the vertex v k−1 . Thus F k 4 has k + 3 vertices and k + 2 edges. Let P k+1 be a path on the vertices v 1 , v 2 , . . . ,v k+1 , where v i is adjacent to v i+1 , i = 1, 2, . . . ,k, k ≥ 1. Let F k 5 be the graph obtained from P k+1 by joining two new vertices to the vertex v k+1 . Note that F k 4 and F k 5 have diameter k + 1. (see Fig. 3). Considering the Lemma 2.4, the follows results are obtained.
Then is an induced subgraph of L r−3 (G). Following this process we concluded is an induced subgraph of G.

Energy of iterated graphs
In this section, we derive bounds on the incidence energy of iterated total graphs of regular graphs. Futhermore, we construct new families of nonisomorphic cospectral graphs.

Incidence energy of iterated graphs
The basic properties of iterated line graph sequences are summarized in the articles [2,4].
The line graph of a regular graph is a regular graph. In particular, the line graph of a regular graph of order n 0 and degree r 0 is a regular graph of order n 1 = 1 2 r 0 n 0 and degree r 1 = 2r 0 − 2.
Consequently, the order and degree of L k (G) are (see [2,4]): n k = 1 2 r k−1 n k−1 and r k = 2r k−1 − 2 where n k−1 and r k−1 stand for the order and degree of L k−1 (G). Therefore, Theorem 3.1 Let G be a regular graph of order n 0 and degree r 0 , then for k ≥ 1 the k-th iterated total graph of G is a regular graph of degree r k and order n k , where 1. r k = 2r k−1 , and 2. n k = n k−1 Proof. Let k ≥ 1. Suppose that the (k − 1)-th iterated total graph of G is a regular graph of order n k−1 and degree r k−1 .
1. Let v be a vertex of the k-th iterated total graph of G then • Case a) If v is a vertex of the (k −1)-th iterated total graph of G then v is adjacent to r k−1 vertices and incident to r k−1 edges in T k−1 (G). Thus, the degree of v in T k (G) is If v is an edge of the (k − 1)-th iterated total graph of G then v is adjacent in each extreme to r k−1 − 1 edges and incident to its two extreme vertices in T k−1 (G).
Thus, the degree of v in T k (G) is Therefore, the k-th iterated total graph of G is a regular graph of degree r k = 2r k−1 . . Therefore, the order of the k-th iterated total graph of G is

Let
Repeated application of the previous theorem generates the following result.
Corollary 3.1 Let G be a regular graph of order n 0 and degree r 0 , then for k ≥ 1 the k-th iterated total graph of G is a regular graph of degree r k and order n k , where and For the next result, we need the following Lemma seen in [9].
Lemma 3.1 [9] Let G be a regular graph of order n and degree r. Then the eigenvalues of T (G) where λ i is an eigenvalue of G.
Now we consider bounds for the incidence energy of iterated total graph. Theorem 3.2 Let G be a regular graph of order n and degree r. Then Equality hold if and only if G ∼ = K 2 .
Proof. 1. Let r = 1 then G is union disjoint of copies of K 2 and T (G) is union disjoint of copies of K 3 . This is, G ∼ = n 2 K 2 and T (G) ∼ = n 2 K 3 , where n is even. Since IE(K 3 ) = 4 it follow that IE(T (G)) = 2n. Therefore, if r = 1 with equality if and only if n = 2, this is, Let r ≥ 2. Since G is a regular graph of degree r. From Theorem 3.1, T (G) is a regular graph of degree 2r. From Lemma 3.1, the signless Laplacian eigenvalues of T (G) are where λ i is an eigenvalue of G.
Equality hold if and only if G ∼ = K 2 and k = 1.

An application: Constructing nonisomorphic cospectral graphs
Many constructions of cospectral graphs are known. Most constructions from before 1988 can be found in [7], §6.1, and [8], §1.3; see also [10], §4.6. More recent constructions of cospectral graphs are presented by Seress [22], who gives an infinite family of cospectral 8-regular graphs. Graphs cospectral to distance-regular graphs can be found in [1,7,8,10,17,22,23]. Notice that the graphs mentioned are regular, so they are cospectral with respect to any generalized adjacency matrix, which in this case includes the Laplace matrix.
Theorem 3.3 Let G 1 and G 2 be two regular graphs of the same order and degree n 0 and r 0 ≥ 3, respectively. Then, for any k ≥ 1 the following hold (a) T k (G 1 ) and T k (G 2 ) are of the same order, and have the same number of edges.
(b) T k (G 1 ) and T k (G 2 ) are cospectral if and only if G 1 and G 2 are cospectral.
Proof. Statement (a) follows from Eqs.