Hyperbolicity on Graph Operators

A graph operator is a mapping F : Γ → Γ′, where Γ and Γ′ are families of graphs. The different kinds of graph operators are an important topic in Discrete Mathematics and its applications. The symmetry of this operations allows us to prove inequalities relating the hyperbolicity constants of a graph G and its graph operators: line graph, Λ(G); subdivision graph, S(G); total graph, T(G); and the operators R(G) and Q(G). In particular, we get relationships such as δ(G) ≤ δ(R(G)) ≤ δ(G) + 1/2, δ(Λ(G)) ≤ δ(Q(G)) ≤ δ(Λ(G)) + 1/2, δ(S(G)) ≤ 2δ(R(G)) ≤ δ(S(G)) + 1 and δ(R(G)) − 1/2 ≤ δ(Λ(G)) ≤ 5δ(R(G)) + 5/2 for every graph which is not a tree. Moreover, we also derive some inequalities for the Gromov product and the Gromov product restricted to vertices.


Introduction
In [1], J. Krausz introduced the concept of graph operators. These operators have applications in studies of graph dynamics (see [2,3]) and topological indices (see [4][5][6]). Many large graphs can be obtained by applying graph operators on smaller ones, thus some of their properties are strongly related. Motivated by the above works, we study here the hyperbolicity constant of several graph operators.
Along this paper, we denote by G = (V(G), E(G)) a connected simple graph with edges of length 1 (unless edge lengths are explicitly given) and V = ∅. 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, Λ(G), is the graph constructed from G with vertices the set of edges of G, and and two 19 vertices are adjacent if and only if their corresponding edges are incident in G.
The subdivision graph, S(G), is the graph constructed from G substituting each of its edges by a path of length 2.
The graph Q(G) is the graph constructed from S(G) byadding edges between adjacent vertices in Λ(G).
The graph R(G) is constructed from S(G) by adding edges between adjacent vertices in G.
The total graph, T(G),is constructed from S(G) by adding edges between adjacent vertices in G or Λ(G).

Main Results
The following result is immediate from the definition of S(G).
Corollary 1. Let G be a graph. Then Proof. Note that S(G) can be considered as a bipartite graph, where V(S(G)) = V(G) ∪ V(Λ(G)).
Since G is an isometric subgraph of T(G) and R(G), and Λ(G) is an isometric subgraph of T(G) and Q(G), we have the following consequence of Lemma 1.

Corollary 2.
For any graph G, we have The hyperbolicity of the line graph has been studied previously (see [21][22][23]). We have the following results. Theorem 2. [22] (Corollary 3.12) Let G be a graph. Then Furthermore, the first inequality is sharp: the equality is attained by every cycle graph.

Proof. Proposition 2 and Theorem 3 give
From Proposition 1, and Theorems 2 and 4 we have: Corollary 2 and Theorems 2, 3 and 4 have the following consequence.

Corollary 4. Let G be a graph. Then
Theorem 4 improves the inequality δ * (Λ(G)) ≤ δ * (G) + 6 in [23]. Given a graph G with multiple edges, we define the graph B(G), obtained from G, substituting each multiple edge for one of its simple edges of shorter length (see [23]).
Proof. Note that R(G) can be obtained by adding an edge of length 2 to each pair of adjacent vertices in G, so the graph becomes a graph with multiple edges, with j = 1 and J = 2. Then [24] (Theorem 8) and Remark 1 give the result.

Lemma 2.
Given the following graphs with edges of length 1, we have • If P n is a path graph, then δ(P n ) = 0 for all n ≥ 1.
• If C n is a cycle graph, then δ(C n ) = n/4 for all n ≥ 3.
• If K n is a complete graph, then δ( If G is not a tree, we define its girth g(G) by From [26] (Theorem 17), we have:

Corollary 6.
If G is not a tree, then

Corollary 7.
If G is not a tree, then Proof. Since G is not a tree, Corollary 6 gives δ(G) ≥ 3/4, and so max δ(G), and Corollary 5 gives the inequalities.
Theorem 2 and Corollary 7 have the following consequence.
From Proposition 1 and Corollary 7 we have the following result.

Theorem 6. Let G be a graph. Then
Proof. The lower bounds follow from Corollary 2. We consider the map P : for i, j ∈ {1, 2, 3}. Thus, Therefore, These inequalities allow us to obtain the result for upper bounds of δ * (Q(G)) and δ * v (Q(G)). The other upper bounds can be obtained similarly.
Let G be a graph, a family of subgraphs {G s } s of G is a T-decomposition if ∪ s G s = G and G s ∩ G r is either a cut-vertex or the empty set for each s = r (see [25]).

Lemma 3. Given a graph G and {G s } s any T-decomposition of G, then
The following results improve the inequality δ(Q(G)) ≤ 6δ(Λ(G)) + 18 in Corollary 11.
Consider the T-decomposition {G n } of Q(G). Since each connected component G n is either a cycle C 3 or a path of length 1, we have δ(Q(G)) = sup n {δ(G n )} ≤ 3/4, by Lemmas 2 and 3.
The union of the set of the midpoints of the edges of a graph G and the set of vertices, V(G), will be denote by N(G). Let T 1 be the set of geodesic triangles T in G such that every vertex of T belong to N(G) and δ 1 (G) := inf{λ : every triangle in T 1 is λ-thin}. The previous lemma allows to reduce the study of the hyperbolicity constant of a graph G to study only the geodetic triangles of G, whose vertices are vertices of G (i.e., belong to V(G)) or midpoints of the edges of G.
For each v ∈ V(G), let us define V v := {u ∈ V(Q(G)) : uv ∈ E(Q(G))} = {u ∈ V(Λ(G)) : uv ∈ E(Q(G))}. Denote by G v and G * v the subgraphs of Q(G) induced by the sets V v ∪ {v} and V v , respectively. Note that both G v and G * v are complete graphs for every v ∈ V(G), and if G * is a complete graph with r vertices, then G v is a complete graph with r + 1 vertices. Also, Q(G) = Λ(G) ∪ (∪ v∈V(G) G v ). By Lemma 4 there exists a geodesic triangle T ∈ T 1 in Q(G) with δ(T) = δ(Q(G)). Denote by γ 1 , γ 2 , γ 3 the sides of T. Without loss of generality we can assume that there exists p ∈ γ 1 with d Q(G) (p, γ 2 ∪ γ 3 ) = δ(T) = δ(Q(G)). Thus, T is a cycle and each vertex of T is either the midpoint of some edge of Q(G) or a vertex of Q(G). If , then there exists at least one vertex of T in G v \ Λ(G). In order to form a triangle T * ⊂ Λ(G) from T, we define γ * i := γ i ∩ Λ(G). Note that, for i ∈ {1, 2, 3}, γ * i is a geodesic, since Λ(G) is a isometric subgraph of Q(G). We denote by x i,j the common vertex of γ i and γ j and by u i and u j the other vertices of γ i and γ j respectively.
We consider the following cases: Case A. We assume that exactly one vertex of T belongs to Q(G) \ Λ(G).
By Lemma 4, we have two possibilities: the vertex of T is a vertex of G or a midpoint of an edge in Let v * be the midpoint of the edge x i x j . Let T 1 be the connected component of T \ Λ(G) joining x i and x j . Note that L(T 1 ) = 2. We analyze the two possibilities: Case A1. Assume that x i,j ∈ V(Q(G)). Let us define We are going to prove that σ i and σ j are geodesics in Λ(G). In fact, we prove now that if γ therefore γ j is not a geodesic obtaining the desired contradiction and we conclude d Q(G) (z j , x j ) ≤ d Q(G) (z j , x i ). Hence, σ i is a geodesic in Λ(G). Case A2. There is an edge e ∈ E(Q(G)) \ E(Λ(G)) such that x i,j is the midpoint of e, thus without loss of generality we can assume that e = x i v, and we define σ i := γ * i and σ j := γ * j ∪ x j x i . Thus, σ i is a geodesic in Λ(G).
have the same endpoints and length; therefore, σ j is also a geodesic in Λ(G).
Case B. Assume that there are two vertices of T in some connected component of By Lemma 4, we have two possibilities again: both vertices of T are midpoints of edges or one vertex of T is a vertex of G and the other is a midpoint of an edge.
We can assume that u i , u j ∈ G v \ G * v for some v. We denote by x i (respectively, x j ) the closest point in γ * i (respectively, γ * j ) to u i (respectively, u j ); then x i x j ∈ E(Λ(G)). Let v be the midpoint of the edge x i x j . Let T 2 be the connected component of T \ Λ(G) joining x i and x j . Note that L(T 2 ) = 2.
We analyze the two possibilities again: Case B1. The vertices u i , u j of T are the midpoints of x i v and x j v. Thus, σ i := γ * i , σ j := γ * j and σ k := x i x j are geodesics in Λ(G).
Case B2. Otherwise, we can assume without loss of generality that u j = v and u i is the midpoint of x i v. We have d Q(G) (u i , x j ) = d Q(G) (u i , x i ) + 1 and so, σ i := γ * i and σ j := γ * j ∪ x j x i are geodesics in Λ(G). In this case we define σ k := {x i }.
We believe that our work may motivate the investigation of related open problems such as: (i) the computation of the hyperbolicity constant on geometric graphs; (ii) the analysis of hyperbolicity on the graph operators reported here (i.e., Λ(G), S(G), T(G), R(G) and Q(G)) when applied to geometric graphs; (iii) the study of the hyperbolicity constants of additional graph operators; and (iv) the identification of the properties of graph operations that break or preserve hyperbolicity.