Recent results on hyperbolicity on unitary operators on graphs

For a geodesic metric space X and for x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] in X . The space X is δ - hyperbolic (in Gromov sense) if any side of T is contained in a δ -neighborhood of the union of the two other sides, for every geodesic triangle T in X . If X is hyperbolic, we denote by δ ( X ) the sharp hyperbolicity constant of X , i.e., δ ( X ) := sup { δ ( T ) : T is a geodesic triangle in X } . In this paper, we collect previous results and prove new theorems on the hyperbolic constant of some important unitary operators on graphs


Introduction
Gromov introduced hyperbolic spaces in [18]. These spaces have many interesting applications in computer science and in various fields of mathematics, including geometric group theory, topology, and analysis (see [2,10,17,18,44,47]). In geometric group theory, hyperbolic groups (i.e., groups whose Cayley graphs are hyperbolic) play a central role, as they have many desirable properties, such as finiteness properties, rigidity properties, and algorithmic properties. Hyperbolic spaces also arise in the study of mapping class groups, Teichmuller spaces, and other moduli spaces of geometric objects.
In geometry and topology, hyperbolic spaces are important in the study of manifolds with negative sectional curvature, as they provide a way to approximate such manifolds by discrete models. Hyperbolic spaces also have interesting properties related to their boundaries, such as the Gromov boundary, which can be used to study the large-scale geometry of the space.
The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups.
The relation between graphs and metric spaces is an important point to consider when studying hyperbolicity. In fact, the notion of hyperbolicity was originally introduced in the context of graphs, and later extended to geodesic metric spaces. The relationship between hyperbolicity in these two settings is well understood, and there are many results that establish equivalence between the two notions (see [10]). Understanding the relationship between hyperbolicity in graphs and geodesic metric spaces is important because we can gain insights into the behavior of these structures and develop new techniques for analyzing them. In particular, the hyperbolicity of a geodesic metric space is equivalent to hyperbolicity of a graph related to it [10]. This result indicates that the study of hyperbolicity can be reduced to the analysis of certain graphs associated with the metric space, which may simplify many computations and proofs.
In a geodesic metric space, a geodesic is a curve that minimizes distance between its endpoints, and is equipped with an arc-length parametrization. A metric space is said to be geodesic if every pair of points in the space can be connected by a geodesic. In general, there may be multiple geodesics between two points x and y in a geodesic metric space, but we can use the notation [xy] to denote any one of them; this notation is ambiguous, but it is very convenient. In the case of a graph, we can use the notation uv to denote the edge that connects vertices u and v.
Let X be a geodesic metric space, and let x 1 , x 2 , x 3 ∈ X be three points. The geodesic triangle T with vertices x 1 , x 2 , x 3 is defined to be the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ], and [x 3 x 1 ] in X. We say that X is δ-hyperbolic if for every geodesic triangle T any point on a geodesic between two points in T is within distance δ of the other sides of T . If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ : X is δ-hyperbolic }; if X is not hyperbolic, we define δ(X) = ∞. We say that X is hyperbolic if X is δ-hyperbolic for some δ ≥ 0. If X has connected components {X i } i∈I , then we define δ(X) := sup i∈I δ(X i ), and we say that X is hyperbolic if δ(X) < ∞.
Gromov hyperbolicity can be formulated in various equivalent ways. The book [17] provides several definitions of Gromov hyperbolicity, each with its own strengths and weaknesses. The authors have chosen the above definition based on its deep geometric meaning [17].
Along this work, G = (V, E) = (V (G), E(G)) will denote a simple (without loops and multiple edges) graph (not necessarily connected) such that V = ∅ and every edge has length 1. In order to consider a connected graph G as a geodesic metric space, we identify any edge uv ∈ E(G) with the real interval [0, 1]. Hence, the points in G are the points in the interior of any edge in E(G) and the vertices in V (G). In this way, any connected graph G has a natural distance defined as the length of the shortest path connecting two points in the graph, and we can see G as a metric graph. We denote by d G or d this distance. If two points are in different connected components of the graph, their distance is defined to be infinity. With this construction, any connected component of a graph is a geodesic metric space in its own right. This construction allows us to apply geometric concepts and tools from metric spaces to the study of graphs, and vice versa.
The study of hyperbolic graphs is a subject of increasing interest in Discrete Mathematics and its applications, for example, networks and algorithms (see [27]), random graphs (see, [40][41][42]), etc. In fact, many real networks are hyperbolic (see [1,28,31,45]). Hyperbolicity in graphs has also been used in issues such as the secure transmission of information through the network (see [23], [25]). Other problems that have been addressed are sensor networks, distance estimation, traffic flow, congestion minimization (see [3,24]). The hyperbolicity constant has been successfully applied to the study of chemical structures, (see [33]) and DNA study (see [11]).
The study of hyperbolic graphs, from a mathematical point of view, has three main objectives: 1.− Study the hyperbolicity constant of some kinds of graphs.
2.− Obtain relationships between the hyperbolicity constant and other parameters of a graph.

3.− Study the invariance of the hyperbolicity constant under transformations.
In this work, in order to achieve the stated objectives, we collect the previous results and prove new theorems on the hyperbolic constant of some important unitary operators on graphs.

Hyperbolicity on unitary operators
In [26], J. Krausz introduced the concept of graph operators. A graph operator is a mapping F : Γ → Γ , where Γ and Γ are families of graphs. Different kinds of graph operators have been investigated in the studies on graph dynamics (see [19,32]) and topological indices (see [9,15,16,34,35,48]). Some large graphs are composed from some existing smaller ones by using graph operators, and many properties of such large graphs are strongly associated with that of the corresponding smaller ones. Motivated from the above works, we study here the hyperbolicity constant of some graph operators.
Given an edge e = uv ∈ E(G) with endpoints u and v, we write V (e) = {u, v}. Next, we recall the definition of some of the main graph operators.
The line graph, denoted by L(G), is the graph whose vertices correspond to the edges of G with two vertices being adjacent if and only if the corresponding edges in G have a vertex in common.
The complement of a graph G, denoted by G, is the graph whose vertices correspond to V (G) but whose edge set consists of the edges not present in G.
The subdivision graph, denoted by S(G), is the graph obtained from G by replacing each of its edge by a path of length two, or equivalently, by inserting an additional vertex into each edge of G.
The para-line graph of G, denoted by P(G), is the line graph of the subdivision graph of G, i.e. P(G) = L(S(G)).
The total graph, denoted by T (G), has as its vertices the edges and vertices of G. Adjacency in T (G) is defined as adjacency or incidence for the corresponding elements of G.
The graph R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge. Another way to describe R(G) is to replace each edge of G by a triangle.
The graph Q(G) is the graph obtained from G by inserting a new vertex into each edge of G and by joining edges those pairs of these new vertices which lie on adjacent edges of G.
The central graph of G, denoted by C(G), is obtained by subdividing every edge of G exactly once and joining all nonadjacent vertices of G in C(G).
Operator R(G) is referred to by some authors as semi-total point or semi-total vertex and operator Q(G) as semi-total line, semi-total edge or middle graph of G (see [4,20,39,46]).
In this section we prove some results relating the hyperbolicity constants of a graph G and its operators S(G), P(G) and C(G). For more information about the R(G), Q(G) and T (G) operators, see [29]. Definition 2.1. Let (X, d) be a metric space and x, y ∈ X. The Gromov product between x and y, with base point w ∈ X, is defined as (x, y) w := 1 2 d(x, w) + d(y, w) − d(x, y) ≥ 0. We say that the metric space (X, d) is δ-hyperbolic with respect to the Gromov product, for some constant δ ≥ 0, if for every x, y, z, w ∈ X.
The following result reported in [2,17] establishes a relationship between the definition given by the Rips condition and the Gromov product.
Theorem 2.1. Let X be a geodesic metric space: 1. If X is δ-hyperbolic with respect to the Gromov product, then X is 3δ-hyperbolic.
2. If X is δ-hyperbolic, then X is 4δ-hyperbolic with respect to the Gromov product.
Let us consider now the hyperbolicity constant with respect to the Gromov product (see Definition 2.1). We denote by δ * (G) the sharp constant for the inequality (1), i.e., In [43] we found the following improvement of the previous inequality: the constant of hyperbolicity of the Gromov product restricted to the vertices of G, i.e., The following result given in [29] relates δ * (G) and δ * v (G).

Subdivision and para-line operators
The following result is immediate from the definition of S(G).
The hyperbolicity of the line graph has been studied previously (see [12][13][14]). The line graph of G is interesting in the theory of geometric graphs, since it is the intersection graph of E(G).
The following result follows from Proposition 2.2 and Theorem 2.5 Note that Theorem 2.5 improves the inequality δ * (L(G)) ≤ δ * (G) + 6 reported in [13]. Given a graph G, we define The following result appears in [38].
From [38] we have the following result.
Theorem 2.7. The following graphs with edges of length 1 have these precise values of δ.
• The path graphs P n satisfies δ(P n ) = 0 for every n ≥ 1.
• The cycle graphs C n satisfies δ(C n ) = n/4 for every n ≥ 3.
Let us denote by V (E) the set of vertices generated by subdividing the edges of G.
Theorem 2.8. The following graphs have the following values of δ: i. The path graph satisfies δ(P(P n )) = 0 for every n ≥ 1.
Proof. If e = v i v j ∈ E(G), then we denote by v vivj the vertex in L(G) corresponding to e.
iii. Note that P(K n ) has n clique subgraphs of (n − 1) vertices associated with the n vertices of G and diam V (P(K n )) = 3. Let x and y be the midpoints of v v4v 3,4 v v4v 1,4 and v v2v 1,2 v v2v 2,3 , respectively. Consider P * and P two geodesics joining x and 3 , v v2v 2,3 }. Let z and p be the midpoints of P * and P , respectively, and consider the geodesic triangle iv. Note that P(S n ) is the complete graph K n−1 with a leaf at each vertex.

Semi total and total operators
If H is a subgraph of G, we always have y) for every x, y ∈ H. Note that this condition is equivalent to d H (u, v) = d G (u, v) for every vertices u, v ∈ V (H). The following result appeared in [38].

Lemma 2.1. If H is an isometric subgraph of G, then δ(H) ≤ δ(G).
Since G is an isometric subgraph of T (G) and R(G), and L(G) is an isometric subgraph of T (G) and Q(G), we have the following consequence of Lemma 2.1.   Given a graph G with multiple edges, we define B(G) as the graph (without multiple edges) obtained from G by replacing each multiple edge by a single edge with the minimum length of the edges corresponding to that multiple edge. From [7] we have the following result. Given any graph G which is not a tree, we define its girth g(G) as the infimum of the lengths of the cycles in G. From [30] we have the following result.

Proposition 2.8. [29, Theorem 7] If G is a path graph, then
The union of the set of the midpoints of the edges of a graph G and the set of vertices, V (G), will be denoted by J(G). Let T 1 be the set of geodesic triangles T in G that are cycles and such that each vertex of T belongs to J(G). Let us define δ 1 (G) := inf{λ : every geodesic triangle in T 1 is λ-thin}.

Central graph
The central graph of G, C(G), has the same set of vertices as S(G), and the set of edges of C(G) is the union of the edge sets of S(G) and G. We can write The central graph is related to the complement graph; in such a direction, it is important to note that hyperbolicity in the complement graph has been studied in [8,22,36,37]. Next, we expose some of its main results. The following result in [8] gives a sharp bound for the hyperbolicity constant of the complement of a graph. Theorem 2.14. If G is a graph with diam V (G) ≥ 3, then its complement graph G satisfies 0 ≤ δ(G) ≤ 2.
The next theorem about regular graphs can be proved using the following result (see [5]). Since G is a k-regular graph with k < n − 2, then G is (n − k − 1)-regular graph. Therefore, G is not a tree and G has a cycle of length at least 3. The result follows because δ(G) ≥ g(G) 4 .
We can write

Theorem 2.17. If G = (V, E) is a graph, then diam V (C(G)) ≤ 4 and this bound is sharp.
Proof. Fix two vertices u, v ∈ V (C(G)). We have three cases: . We have the following cases: The bound is attained, for example, on the central graph of the complete graph K n (n ≥ 4) by considering the vertices obtained by the subdivision of two non-adjacent edges in K n .
Proof. Let x, y ∈ C(G) be points such that d C(G) (x, y) = diam C(G). Since the diameter of the graph is reached by considering the distances between vertices and midpoints of edges, let us consider the following cases: (i) x, y ∈ V (C(G)). Theorem 2.17 gives d C(G) (x, y) ≤ 4.
(iii) x, y are the midpoints of u 1 u 2 , v 1 v 2 ∈ E(C(G)), respectively. Without loss of generality we can assume that u 1 , v 1 ∈ V (G). Case 1 in Theorem 2.17 gives Note that it is not necessarily the case that diam V (C(G)) = diam C(G). For example, if G = P 2 , then C(P 2 ) = C 5 and diam V (C 5 ) = 2 = 5/2 = diam C(G). Theorem 2.6 and Corollary 2.10 give the next result.
Theorem 2.19. If G is a graph of order n ≥ 2, then the following statements hold: ii. diam V (C(G)) = 2 if and only if G = S n .
iii. diam V (C(G)) = 4 if and only if G has an induced subgraph K 4 .
Proof. i. If diam V (C(G)) = 1, then for each pair of vertices u, v ∈ V (G), we have that uv ∈ E(G) and so, E(G) = ∅. The converse is immediate.
ii. Assume now that diam V (C(G)) = 2. If G has an induced subgraph If the length of the girth of G is greater than or equal to 4, then there exist two non-adjacent edges u 1 u 2 and v 1 v 2 , and d C(G) (u 1,2 , v 1,2 ) = 3. Therefore, G has no cycles, i.e., G is a tree. Since diam V (C(G)) = 2, then each pair of edges in E(G) has a common vertex in V (G) and so, G = S n . The converse is immediate.
iv. By the previous item, we have and so, diam C(G) = 3. v. Corollary 2.10 gives the result.
Theorem 2.7 gives the following result.

Theorem 2.21.
If G is a graph of order n ≤ 3, then the following statements hold: i. If n ≤ 2, then δ(C(G)) = 0 .
ii. If G is a set of three isolated vertices, then δ(C(G)) = 3/4.
iii. If G is an edge and an isolated vertex, then δ(C(G)) = 1.

Theorem 2.22.
If G is a graph of order n ≥ 4, then Proof. Since Theorem 2.20 relates diam C(G) and diam V (C(G)), we have the following cases. Case i. diam V (C(G)) = 1. By the argument in the proof of Theorem 2.20, we have C(G) = K n and diam C(G) = 2, and Theorem 2.7 gives δ(C(G)) = 1 = 1 2 diam C(G).
Let x, y ∈ C(G) be the midpoints of v 1 v 2 and v 0 v 0,3 , respectively. Note that d C(G) (x, y) = 3 = diam C(G). Let P and P be two geodesics joining x and y such that P ∩ V (C(G)) = {v 1 , v 3 , v 0,3 } and P ∩ V (C(G)) = {v 0 , v 0,2 , v 2 }. Let z and p be the midpoints of P and P , respectively, and consider the geodesic triangle T = {[xz], [yz], P }. We have Case iii. diam V (C(G)) = 3. We have two cases.
Case iii-A. G contains either H 0 or H 1 as subgraphs. Suppose that H 1 ⊂ G. Let x ∈ C(G) be the midpoint of v 3 v 3,4 and y = v 1,2 . Let P and P be geodesics joining x and y such that P ∩ V (C(G)) = {v 1,2 , v 2 , v 4 , v 3,4 } and P ∩ V (C(G)) = {v 1,2 , v 1 , v 1,3 , v 3 }. Let z and p be the midpoints of P and P , respectively, and consider the geodesic trian- Therefore, δ(C(G)) = 1 2 diam C(G). The proof is similar if H 2 ⊂ G. Case iii-B. G does not contain H 0 or H 1 as subgraphs. Theorem 2.6 gives δ(C(G)) ≤ 3/2. Let us prove the converse inequality. Let u, v ∈ V (C(G)) such that d C(G) (u, v) = 3. We have two cases.
If u 1 v 1 ∈ E(G), then u 1 v 2 , u 2 v 1 ∈ E(G). Let P and P be two geodesics joining u and v such that P ∩ V (C(G)) = {u, u 1 , v 2 , v} and P ∩ V (C(G)) = {u, u 2 , v 1 , v}. If u 1 v 1 ∈ E(G), consider P and P such that P ∩ V (C(G)) = {u, u 1 , v 1 , v} and P ∩ V (C(G)) = {u, u 2 , v 2 , v}. Let p and z be the midpoints of P and P , respectively, and consider the geodesic triangle   • The path graph P n satisfies δ(C(P n )) = 3/2 for every n ≥ 4.