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Minimum generalized degree distance of n-vertex tricyclic graphs
Journal of Inequalities and Applicationsvolume 2013, Article number: 548 (2013)
In (Hamzeh et al. in Stud. Univ. Babeş-Bolyai, Chem., 4:73-85, 2012), we introduced a generalization of a degree distance of graphs as a new topological index. In this paper, we characterize the n-vertex tricyclic graphs which have the minimum generalized degree distance.
MSC:05C12, 05C35, 05C05.
Topological indices and graph invariants based on the distances between the vertices of a graph are widely used in theoretical chemistry to establish relations between the structure and the properties of molecules. They provide correlations with physical, chemical and thermodynamic parameters of chemical compounds [1–3]. The Wiener index is a well-known topological index, which equals to the sum of distances between all pairs of vertices of a molecular graph . It is used to describe molecular branching and cyclicity and establish correlations with various parameters of chemical compounds. In this paper, we only consider simple and connected graphs. Let G be a connected graph with the vertex and edge sets and , respectively, and the number of vertices and edges of G are denoted, respectively, by n and m. As usual, the distance between the vertices u and v of G is denoted by ( for short). It is defined as the length of any shortest path connecting u and v in G. We let be the degree of a vertex v in G. The eccentricity of v, denoted by , is the maximum distance from vertex v to any other vertex. The diameter of a graph G is denoted by and is the maximum eccentricity over all vertices in a graph G. The radius is defined as the minimum of over all vertices . A connected graph G with n vertices and m edges is called tricyclic if . The join of two graphs and with disjoint vertex sets and and edge sets and is the graph union together with all the edges joining and .
Additively weighted Harary index is defined as follows in 
Generalized degree distance is denoted by and defined as follows in .
For every vertex x, is defined by , where , and to avoid confusion, we denote in graph G with . So we have
where λ is a real number. If , then . When , this new topological index equals the degree distance index (i.e., the Dobrynin or the Schultz index). The properties of the degree distance index were studied in [9–12]. Also, if , then (see above). The relation of our new index with other intensely studied indices motivated our present (and future) study.
Extremal graph theory is a branch of graph theory that studies extremal (maximal or minimal) graphs, which satisfy a certain property. Extremality can be taken with respect to different graph invariants, such as order, size or girth. The problem of determining extremal values and corresponding extremal graphs of some graph invariants is the topic of several papers, for example, see [9–12, 14–16].
2 Main results
In this paper, we characterize all of n-vertex tricyclic graphs which have the minimum generalized degree distance.
Lemma 2.1 
Let and G be an n-vertex tricyclic graph. The integers are the degrees of the vertices of a graph G if and only if
at least four of them are greater than or equal to 2,
for sufficiently large n.
Let be the number of vertices of degree i of G for . If , then
We obtain the minimum of over all integers numbers , which satisfy one of the conditions of Lemma 2.1.
We rewrite Lemma 2.1 in terms of the notations above as follows.
Lemma 2.2 
Let and G be an n-vertex tricyclic graph. The integers are the multiplicities of the degrees of a graph G if and only if
for sufficiently large n.
We denote the set of all vectors , which satisfy the conditions above, by Δ. Let G be a connected graph with multiplicities of the degrees , and let , , , and . Now we consider the transformation of , which is defined as follows :
We have for and , , , .
Let , . Now consider the transformation defined as follows :
That is for and , , .
Lemma 2.3 Suppose that λ is a positive integer number, and consider the set of vectors .
If , then , unless and ,
Proof (1) We can easily see that , . If , , then , and in this case, . Now if , according to , we have and . Therefore, we conclude that if , then if and only if and .
With a simple calculation, we have
The proof is now completed. □
Lemma 2.4 Suppose that λ is a positive integer number, and consider the set of vectors .
If , then , unless and ,
Proof The proof is similar to the proof of the previous lemma, by taking . □
Theorem 2.5 Let and λ be a positive integer. If G belongs to the class of connected tricyclic graphs on n vertices, then we have
and all the extremal graphs are isomorphic to H or , where H is obtained by identifying the center of a star with an arbitrary vertex of a complete graph , and is obtained by identifying the center of a star with an arbitrary four degree vertex of a graph , which is shown in Figure 1, respectively.
Proof To find minimum , over the classes of connected tricyclic graphs with n vertices, it is enough to find . At first, we consider the case . In this case, only tricyclic graph with 4 vertices is , that is, , and the theorem is proved in this case.
Now let . Let us consider . The all connected tricyclic graphs with 5 vertices, is , , and . Meanwhile, , , where , , and , respectively, where () are receipted in Figure 1. So and are extremal graphs, and the theorem is proved in this case.
Finally, let . If , consider two different vertices such that . Since , we can choose four different vertices . Actually, the vertices p, q, r, s are all adjacent to x and y, hence G has at least four cycles , , , and , which contradicts the hypothesis. Therefore, .
Let us analyze the possible values for . If there exist such that and , then, by applying for the positions i and j, we obtain a new vector , for which . Similarly, if there exists such that , then by , we obtain a new degree sequence in Δ, for which , that is, () are not greater than 1. Let us show that all vectors , realizing the minimum of , have .
Actually, suppose that there is an index such that and for all , . In this case, if , we can apply for positions 4 and i and obtain a smaller value for . Suppose that . Since , we will separately analyze the two cases: (a) and (b) .
In this case, , where and . We can consider different vertices such that , , p, q, r, s, all are adjacent to x and y, respectively. Meanwhile, x and y are adjacent. We have found four cycles , , , and , which contradicts the hypothesis.
If , then , (), and Δ is characterized by the equations , , and . Suppose that . In this case, we can use for position 3 and i and deduce a smaller value for . Thus and . We can obtain . Moreover, . In this case, we can use for position 2 and i and deduce a smaller value for .
To sum up, we have and . We will separately analyze the two cases: (a′) and (b′) again.
(a′) If , then . This equation does not hold. If all , , and are not greater than 2, , which contradicts the hypothesis . If one of , , and is greater than 2, then by using for the corresponding position, we can obtain a smaller value for .
(b′) If , then and , namely . If , then . If , then and , which contradicts the condition (iii) in Lemma 2.2. Thus, or . We now distinguish the following two subcases:
Subcase 1. If , then . All its solutions are , or , or , or , . In the third and fourth cases, by using for the position 2, we can obtain a smaller value for . In the second case, by using for the position 2, we obtain the first case. But the first case cannot be changed, since by using for the position 3, the vector can be , which contradicts Lemma 2.2. Thus, , and the corresponding unique graph is denoted by H.
Subcase 2. If , then . All its solutions are , or , . The first case should be removed. By a similar reasoning as before, the second case cannot be transferred by . Meanwhile, the first transformation cannot be used. Otherwise, the sequence can be changed into , which contradicts Lemma 2.2. Thus, , and the corresponding unique graph is denoted by .
We can easily obtain and . Thus, . So, case (b) holds. This completes the proof of Theorem 2.5. □
In , the authors proved the following theorem.
Theorem 2.6 Let . If G belongs to the class of connected tricyclic graphs on n vertices, then:
If , then and the unique extremal graph is isomorphic to ;
If , then , and all the extremal graphs are isomorphic to H or , where H is obtained by identifying the center of a star with an arbitrary vertex of a complete graph , and is obtained by identifying the center of a star with an arbitrary degree 4 vertex of a graph , which is shown in Figure 1, respectively.
If we choose in Theorem 2.5, then we can obtain the same result (Theorem 3.1 in ).
Theorem 2.7 Let G belong to the class of connected graphs on n vertices, and let λ be a positive integer number, then for every , we have
and the unique extremal graph is .
Proof Since for the property is trivial, so let . If G is a connected graph with n vertices, then and .
We will find the minimum of over all natural numbers satisfying and . First, we shall prove that under these conditions, all systems of natural numbers reaching must satisfy (i) and (ii) given below:
Indeed, suppose that . It follows that there exists a smallest index m, such that . We shall define the system such that , and for every . We have , and .
Hence cannot be minimum for systems , for which . Since natural numbers are the degrees of the vertices of a tree of order n if and only if , we can consider that the numbers are the multiplicities of the degrees of the vertices of a tree T of order n, and we shall denote also instead of . Hence the domain where must be minimized is defined by the multiplicities of the degrees of the vertices of a tree T of order n.
If T is not isomorphic to , i.e., if , then . Let such that . It follows that , and denote by u and v, , the vertices adjacent with t and z, respectively. Suppose that and , where and .
Let be the tree of order n defined in the following way: and . By replacing T with , the degrees of u and v change: , , and the degrees of vertices remain unchanged. We get , and cannot be minimum.
It follows that is minimum only if , i.e., , and , when . By concluding, , and equality holds if and only if , and , i.e., since . □
In , the authors proved the following theorem.
Theorem 2.8 Let G belong to the class of connected graphs on n vertices, then for every , we have
and the unique extremal graph is .
If we choose in Theorem 2.7, then we can obtain the same result (Theorem 2.1 in ).
A Moore graph is a graph of diameter k with girth . Those graphs have the minimum number of vertices possible for a regular graph with given diameter and the maximum degree. We first bring the following results in [20, 21].
Lemma 2.9 
Let G be a connected graph of order and size , and let λ be a negative integer. Then , and the equality holds if and only if , where d is the diameter of G.
Lemma 2.10 
Let G be a triangle- and quadrangle-free graph on n vertices with radius r. Then . The equality is valid if and only if G is a Moore graph of diameter 2 or .
Theorem 2.11 Let G be a triangle- and quadrangle-free graph on n vertices and m edges with radius r, and let λ be a negative integer. Then . The equality is valid if and only if G is a Moore graph of diameter 2 or .
Proof By the lemmas above, we have
The first equality holds if and only if the diameter of G is at most 2, and the second one holds if and only if G is a Moore graph of diameter 2 or . So, the equality holds if and only if G is a Moore graph of diameter 2 or . This completes the proof. □
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The authors would like to thank the referee for the valuable comments.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.