], J. Krausz introduced the concept of graph operators. These operators have applications in studies of graph dynamics (see [2
]) and topological indices (see [4
]). 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 a connected simple graph with edges of length 1 (unless edge lengths are explicitly given) and . Given an edge with endpoints u and v, we write . Next, we recall the definition of some of the main graph operators.
The line graph, , 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, , is the graph constructed from G substituting each of its edges by a path of length 2.
The graph is the graph constructed from byadding edges between adjacent vertices in .
The graph is constructed from by adding edges between adjacent vertices in G.
The total graph, ,is constructed from by adding edges between adjacent vertices in G or .
So, we have the following:
The Gromov hyperbolic spaces have multiple applications both theoretical and practical (see [7
]). A space is geodesic if any two points in it can be joined by a curve whose length is the distance between them. In this paper we will consider a graph G
as a geodesic metric space and any geodesic joining x
will be denote by
Let X be a geodesic metric space and . A geodesic triangle with vertices , denoted by , is the union of three geodesics , and . We write also . If the -neighborhood of the union of any two sides of T contains the other side, we say that T is -thin. We define The space X is -hyperbolic if all geodesic triangles T in X are -thin. Let us denote the sharp hyperbolicity constant of X, by , i.e., X is Gromov hyperbolic if X is -hyperbolic for some ; then X is Gromov hyperbolic if and only if .
In this paper we prove inequalities relating the hyperbolicity constants of a graph G and its graph operators , , , and , using their symmetries.
3. Main Results
The following result is immediate from the definition of .
Proposition 1. Let G be a graph. Then
We remark that the equality is not true for
), but inequalities may apply. The next result appears in [21
Theorem 1. Let be a bipartite graph. We have , where
for every .
Corollary 1. Let G be a graph. Then
Note that can be considered as a bipartite graph, where . Theorem 1 gives Since , the desired inequalities hold. ☐
Proposition 2. Let G be a graph. Then
The inequality is direct. Let us prove the other inequality.
and we conclude
Let H be a subgraph of G, H is isometric if for every . We will need the following well-known result.
Lemma 1. Let H be an isometric subgraph of G. Then
Since G is an isometric subgraph of and , and is an isometric subgraph of and , 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
]). We have the following results.
Theorem 2.  (Corollary 3.12) Let G be a graph. Then
Furthermore, the first inequality is sharp: the equality is attained by every cycle graph.
Theorem 3.  (Theorem 6) Let G be a graph. Then
Theorem 4. Let G be a graph. Then
Proposition 2 and Theorem 3 give , and . ☐
From Proposition 1, and Theorems 2 and 4 we have:
Corollary 3. Let G be a graph. Then
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
Given a graph G
with multiple edges, we define the graph
, obtained from G
, substituting each multiple edge for one of its simple edges of shorter length (see [23
Remark 1. By argument in the proof of  (Theorem 8) we have: If in each multiple edge there is at most one edge with length greater than , then , where, . Corollary 5. Let G be a graph. Then
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
. Then [24
] (Theorem 8) and Remark 1 give the result. ☐
] (Theorem 11), we have the following result.
Given the following graphs with edges of length 1, we have
If is apath graph, then for all
If is acycle graph, then for all
If is acomplete graph, then , and for all
is not a tree, we define its girth
] (Theorem 17), we have:
Corollary 7. If G is not a tree, then
is not a tree, Corollary 6 gives
, and so
and Corollary 5 gives the inequalities. ☐
Theorem 2 and Corollary 7 have the following consequence.
Corollary 8. If G is not a tree, then
From Proposition 1 and Corollary 7 we have the following result.
Corollary 9. If G is not a tree, then
Theorem 6. Let G be a graph. Then
The lower bounds follow from Corollary 2. We consider the map
is an isometric subgraph of
These inequalities allow us to obtain the result for upper bounds of and The other upper bounds can be obtained similarly. ☐
From Theorems 3 and 6 and Corollary 4 we have:
Corollary 10. For all graph G, we have
From Corollaries 2, 4 and 10, Theorem 6 and the inequalities and , we have:
Corollary 11. Let G be a graph. Then
Corollaries 2 and 4 give the lower bounds. On the other hand, Theorem 6 gives , ; we obtain the third upper bound in a similar way. Corollary 10 gives , obtaining the last upper bound. ☐
be a graph, a family of subgraphs
is a T-decomposition
is either a cut-vertex
or the empty set for each
The following result was proved in [24
] (Theorem 3).
Lemma 3. Given a graph G and any T-decomposition of G, then
The following results improve the inequality in Corollary 11.
Theorem 7. Let G be a path graph, then
Since G is a path graph, is also a path graph, and so .
Consider the T-decomposition of . Since each connected component is either a cycle or a path of length 1, we have , 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, , will be denote by . Let be the set of geodesic triangles T in G such that every vertex of T belong to and .
Lemma 4.  (Theorems 2.5 and 2.7) For every graph G, we have . Furthermore, if G is hyperbolic, then there exists with .
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 ) or midpoints of the edges of G.
Theorem 8. If G is not a path graph, then
By Corollary 2 we have the first inequality. We will prove the second one. If , then Theorem 6 gives , and the second inequality holds. Assume now that (and so, by Theorem 6). If G is not a path, then is not a tree and Corollary 6 gives .
For each , let us define = . Denote by and the subgraphs of induced by the sets and , respectively. Note that both and are complete graphs for every , and if is a complete graph with r vertices, then is a complete graph with vertices. Also,
By Lemma 4 there exists a geodesic triangle in with . Denote by the sides of T. Without loss of generality we can assume that there exists with . Thus, T is a cycle and each vertex of T is either the midpoint of some edge of or a vertex of .
If contains to T for some , then by Lemma 2, since is an isometric subgraph of .
If contains to T, then by Lemma 1, since is isometric.
Suppose that T is not contained either in nor with .
Note that if for some , then there exists at least one vertex of T in . In order to form a triangle from T, we define . Note that, for , is a geodesic, since is a isometric subgraph of .
We denote by the common vertex of and and by and the other vertices of and respectively.
We consider the following cases:
Case A. We assume that exactly one vertex of T belongs to . Thus, there exists such that . By Lemma 4, we have two possibilities: the vertex of T is a vertex of G or a midpoint of an edge in .
We can suppose that Let v be a vertex of such that . Let (respectively, ) be the closest point of (respectively, ) to . Thus, . Let be the midpoint of the edge . Let be the connected component of joining and . Note that . We analyze the two possibilities:
Case A1. Assume that
. Let us define
We are going to prove that
are geodesics in
. In fact, we prove now that if
. Seeking for a contradiction assume that
is not a geodesic obtaining the desired contradiction and we conclude
is a geodesic in
Case A2. There is an edge such that is the midpoint of e, thus without loss of generality we can assume that , and we define and Thus, is a geodesic in
Note that and have the same endpoints and length; therefore, is also a geodesic in .
Case B. Assume that there are two vertices of T in some connected component of . Thus, there exists such that . 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 for some v. We denote by (respectively, ) the closest point in (respectively, ) to (respectively, ); then . Let be the midpoint of the edge . Let be the connected component of joining and . Note that .
We analyze the two possibilities again:
Case B1. The vertices of T are the midpoints of and . Thus, , and are geodesics in .
Case B2. Otherwise, we can assume without loss of generality that and is the midpoint of . We have and so, and are geodesics in . In this case we define .
Note that the most general possible case is the following: there are at most three vertices such that , for . Repeating the previous process at most three times we obtain a geodesic triangle in with sides , and containing , and , respectively.
If , then one can check that . If , then ; since , we have . This finishes the proof. ☐
Proposition 1, Theorems 2 and 8, and Corollary 3 have the following consequence.
Corollary 12. Let G be a graph. If G is not a path graph, then