Abstract
The reformulated reciprocal degree distance is defined for a connected graph as , which can be viewed as a weight version of the Harary index; that is, . In this paper, we present the reciprocal degree distance index of the complement of Mycielskian graph and generalize the corresponding results to the generalized Mycielskian graph.
1. Introduction
Throughout this paper we consider (nontrivial) simple graphs, which are finite and undirected graphs without loops or multiple edges. Let be a connected graph. For vertices , the distance between and in , denoted by , is the length of a shortest path in and let be the degree of a vertex . A chemical graph is a graph whose vertices denote atoms and edges denote bonds between those atoms of the underlying chemical structure. A topological index for a (chemical) graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In theoretical chemistry, molecular structure descriptors (also called topological indices) and graph invariants based on the distances between vertices of a graph or vertex degree are widely used for characterizing molecular graphs, establishing relations between structure and properties of molecules, predicting biological activity of chemical compounds, and making their chemical applications.
The Wiener index of is defined as with the summation going over all pairs of distinct vertices of . Similarly, the Harary index of is defined as .
Das et al. [1] consider the generalized version of Harary index, namely, the Harary index, which is defined as . Its applications and mathematical properties are well studied in [2–5].
Dobrynin and Kochetova [6] and Gutman [7] independently proposed a vertex-degree-weighted version of Wiener index called degree distance, which is defined for a connected graph as . Note that the degree distance is a degree-weight version of the Wiener index. Hua and Zhang [8] introduced a new graph invariant named reciprocal degree distance, which can be as a degree-weight version of Harary index; that is, . The reciprocal degree distance of some graph operations is obtained in [9, 10].
Recently, Li et al. [11] introduced a vertex-degree-weighted version of Harary index called reformulated reciprocal degree distance, which is defined for a connected graph as . In view of is just the additively weighted Harary index, while in view of it is also the generalized version of reciprocal degree distance of a connected graph . The mathematical properties of the three edge-grafting transformations and some sharp upper bounds on the reformulated reciprocal degree distance of trees are presented in [11].
The first Zagreb index is defined as and the second Zagreb index is defined as . The Zagreb indices are found to have applications in QSPR and QSAR studies as well; see [12].
Mycielski [13] developed a graph transformation that transforms a graph into a new graph , which is called the Mycielskian of , in a search for triangle-free graphs with arbitrarily large chromatic number. The generalized Mycielskian is natural generalization of Mycielski graphs, which is also called by Tardif cones over graphs [14]. Let be a graph with vertex set and edge set . Given an integer , the Mycielskian of , denoted by , is the graph with vertex set , where is the th distinct copy of for and edge set . Concerning on the other results for (generalized) Mycielskian graph, one can refer to papers [15–19].
2. Complement of the Mycielskian Graph
In this section, we obtain the reformulated reciprocal degree distance and reciprocal degree distance of the graph .
The following lemmas follow from the structure of the complement of the Mycielskian graph.
Lemma 1. Let be a connected graph. Then the distances between the vertices of the Mycielskian graph of are given as follows. For each ,
Lemma 2. Let be a graph on vertices. Then the degree of any vertex
Remark 3. Let be a graph with . Then there are two element subsets in . Therefore
Theorem 4 (corresponding to Theorem 3.4 in [20]). Let be a graph on vertices and edges with diameter . Then .
Proof. From the structure of the complement of Mycielskian graph, we consider the following cases of adjacent and nonadjacent pairs of vertices in to compute :
(i) If , then by Lemmas 1 and 2(ii) If , then for each and otherwise. Thereforeby Lemmas 1 and 2(iii) If and , , then by Lemmas 1 and 2(iv) If and , , then by Lemmas 1 and 2Each can be paired with vertices as
(v) If and , then by Lemmas 1 and 2(vi) If and , then by Lemmas 1 and 2Summarizing the total contributions of above cases of adjacent and nonadjacent pairs of vertices in , we can obtain the desired result. This completes the proof.
Using in Theorem 4, we obtain the reciprocal degree distance of the complement of the Mycielskian graph.
Corollary 5 (corresponding to Corollary 3.5 in [20]). Let be a graph on vertices and edges with diameter . Then .
3. Complement of the Generalized Mycielskian Graph
In this section, we obtain the reformulated reciprocal degree distance and reciprocal degree distance of the graph .
The following lemmas follow from the structure of the complement of the generalized Mycielskian graph.
Lemma 6. Let be a connected graph. Then the distances between the vertices of the generalized Mycielskian graph of are given as follows. For each
Lemma 7. Let be a graph with vertices. Then the degree of any vertex is as follows:
Theorem 8. Let be a graph on vertices and edges with diameter . Then .
Proof. From the structure of the complement of Mycielskian graph, we consider the structure of the complement of generalized Mycielskian and then also consider the following cases of adjacent and nonadjacent pairs of vertices in to compute .
Case 1. If ,, then and by Lemmas 6 and 7Case 2. If , , then and by Lemmas 6 and 7Case 3. If , , then and by Lemmas 6 and 7Case 4. If , thenand by Lemmas 6 and 7Case 5. If , , thenand by Lemmas 6 and 7Case 6. If , then and by Lemmas 6 and 7Summarizing the total contributions of above cases of adjacent and nonadjacent pairs of vertices in , we can obtain the desired result; this completes the proof.
Using in Theorem 8, we obtain the reciprocal degree distance of the complement of the generalized Mycielskian graph.
Corollary 9. Let be a graph on vertices and edges with diameter . Then
Data Availability
In this paper, we correct the second part results in the paper [K. Pattabiraman, M. Vijayaragavan, Reformulated Reciprocal Degree Distance of Transformation Graph, Electronic Notes in Discrete Mathematics 53 (2016) 259-270] and generalize the corresponding results to the generalized Mycielskian graph.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by NSFC (Grants nos. 11761070, 61662079, and 11361062), Xinjiang Normal University Undergraduate Teaching Project (SDJG2017-3), the 13th Five-Year Plan for Key Discipline Mathematics of Xinjiang Normal University (no. 17SDKD11), 2017 Education Reform Project of Xinjiang Normal University, and Outstanding Young Teachers Scientific Research Foundation of Xinjiang Normal University (XJNU201416).