Eccentric connectivity index and eccentric distance sum of some graph operations
Let $G=(V,E)$ be a connected graph. The eccentric connectivity index of $G$, $xi^{c}(G)$, is defined as $xi^{c}(G)=sum_{vin V(G)}deg(v)ec(v)$, where $deg(v)$ is the degree of a vertex $v$ and $ec(v)$ is its eccentricity. The eccentric distance sum of $G$ is defined as $xi^{d}(G)=sum_{vin V(G)}ec(v)D...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Isfahan
2013-03-01
|
| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/?_action=showPDF&article=2839&_ob=e28e6997585de2cc61c0727cd04dec50&fileName=full_text.pdf. |
| Summary: | Let $G=(V,E)$ be a connected graph. The eccentric connectivity index of $G$, $xi^{c}(G)$, is defined as $xi^{c}(G)=sum_{vin V(G)}deg(v)ec(v)$, where $deg(v)$ is the degree of a vertex $v$ and $ec(v)$ is its eccentricity. The eccentric distance sum of $G$ is defined as $xi^{d}(G)=sum_{vin V(G)}ec(v)D(v)$, where $D(v)=sum_{uin V(G)}d_{G}(u,v)$ and $d_{G}(u,v)$ is the distance between $u$ and $v$ in $G$. In this paper, we calculate the eccentric connectivity index and eccentric distance sum of generalized hierarchical product of graphs. Moreover, we present explicit formulae for the eccentric connectivity index of $F$-sum graphs in terms of some invariants of the factors. As applications, we present exact formulae for the values of the eccentric connectivity index of some graphs of chemical interest such as $C_{4}$ nanotubes, $C_{4}$ nanotoris and hexagonal chains. |
|---|---|
| ISSN: | 2251-8657 2251-8665 |