The Wiener index of a graph is defined to be the sum of the distances of all pairs of vertices in the graph.The kth power G^(k) of a graph G is the graph on V(G)and two vertices are adjacent if and only if their dista...The Wiener index of a graph is defined to be the sum of the distances of all pairs of vertices in the graph.The kth power G^(k) of a graph G is the graph on V(G)and two vertices are adjacent if and only if their distance in G is less or equal to k.In this paper,we computed the Wiener index of the kth power of paths and cycles for any k≥2.展开更多
According to the definitions of molecular connectivity and hyper-Wiener index, a novel set of hyper-Wiener indexes （Dn, ^mDn） were defined and named as condensed extended hyper-Wiener index, the potential usefulness...According to the definitions of molecular connectivity and hyper-Wiener index, a novel set of hyper-Wiener indexes （Dn, ^mDn） were defined and named as condensed extended hyper-Wiener index, the potential usefulness of which in QSAR/QSPR is evaluated by its correlation with a number of C3-C8 alkanes as well as by a favorable comparison with models based on molecular connectivity index and overall Wiener index.展开更多
The undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) of a finite group <i>Z<sub>n</sub></i> is the graph with vertex set G and two distinct vertices u a...The undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) of a finite group <i>Z<sub>n</sub></i> is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if <i>u</i> ≠ <i>v</i> and <img src="Edit_3b1df203-9ff2-4c13-93d1-4bba568eae54.png" width="40" height="20" alt="" /> or <img src="Edit_094c8f88-deb6-4f41-825a-ba91c0306ae8.png" width="40" height="20" alt="" />. The Wiener index <i>W</i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of an undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be sum <img src="Edit_348337df-b9c2-480d-9713-ec299a6fcd4e.png" width="110" height="25" alt="" /> of distances between all unordered pair of vertices in <i>P</i>(<i>Z<sub>n</sub></i>). Similarly, the edge-Wiener index <i>W<sub>e</sub></i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be the sum <img src="Edit_e9b89765-f71e-4865-a0c5-c688710ff0c6.png" width="60" height="25" alt="" /> of distances between all unordered pairs of edges in <i>P</i>(<i>Z<sub>n</sub></i>). In this paper, we concentrate on the wiener index of a power graph <img src="Edit_dff0cd99-eb11-4123-a437-78cbbd8ebf96.png" width="40" height="20" alt="" />, <i>P</i>(<i>Z<sub>pq</sub></i>) and <i>P</i>(<i>Z<sub>p</sub></i>). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph <i>P</i>(<i>Z<sub>n</sub></i>), using <i>m,n</i> and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of <i>Z<sub>n</sub></i>.展开更多
From a dataset of macrobenthos obtained from 18 cruises from 2004 to 2013 in the Huanghe（Yellow River）Estuary and its adjacent areas,the composition and characteristics of macrobenthos were analyzed,and the applicab...From a dataset of macrobenthos obtained from 18 cruises from 2004 to 2013 in the Huanghe（Yellow River）Estuary and its adjacent areas,the composition and characteristics of macrobenthos were analyzed,and the applicability of the Shannon-Wiener Index（H′）,AZTI＇s Marine Biotic Index（AMBI） and multivariate AMBI（MAMBI） for assessing benthic habitat quality was compared.The results showed a total of 203 macrobenthos in the study area.The macrobenthos were dominated by polychaetes,followed by mollusks and crustaceans.The macrobenthic ecological groups were dominated by EGI,EGII and EGIII,which respectively accounted for 31.5%,36.0% and 21.2% of the total.There were significant differences between the evaluation results of the three indices.The ecological quality status（EQS） levels given by the AMBI were greater than those given by the H′ and M-AMBI.The AMBI could not reflect the differences between 11 sites but the H′ and M-AMBI could do.Moreover,the three indices responded well to the variations in salinity（S） and dissolved oxygen（DO） in the waters.The H′ and MAMBI also responded sensitively to the differences in physical parameters,such as water depth and sediment texture.The correlation between M-AMBI and environmental pressure gradient data was the strongest.The MAMBI could effectively distinguish degraded conditions from undegraded but the H′ and AMBI could not.Therefore,the M-AMBI reflected benthic habitat health well in the study areas.However,the objectivity of evaluation results of M-AMBI needs further verification by physical,chemical and biological methods.The thresholds also need further discussion.展开更多
In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wie...In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. Discrete Appl. Math., 2014, 171：35 41）, and give a simple way to compare the Wiener index and the Zagreb indices for trees. Moreover, the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.展开更多
Based on distance matrixW,the novel topological indexW F is defined by the matricesX,W,L,asW F=WXL.WhereX is a row matrix,which expresses the bonding characteristics between adjacent atoms;L is a column matrix express...Based on distance matrixW,the novel topological indexW F is defined by the matricesX,W,L,asW F=WXL.WhereX is a row matrix,which expresses the bonding characteristics between adjacent atoms;L is a column matrix expressing the characteristic of vertexes in molecules;W is a distance matrix,which expresses the chemical environment of the atoms in the molecule.the properties,the chemical environments and the interaction of the vertexes in a molecule are taken into account in this definition.Hence the application of the indexW F to multiple bonds and heteroatomcontaining organic systems or inorganic systems is possible.Good correlation coefficients are achieved not only in the prediction of the standard formation enthalpy of methyl halides,titanium halides,but also the retention index of gas chromatography of the hydrocarbons.展开更多
The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we character...The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we characterize the graphs with the maximum and second-maximum Wiener indices among all the graphs in G＊c, respectively.展开更多
In this paper, in terms of Wiener index, hyper-Wiener index and Harary index, we first give some sufficient conditions for a nearly balance bipartite graph with given minimum degree to be traceable. Secondly, we estab...In this paper, in terms of Wiener index, hyper-Wiener index and Harary index, we first give some sufficient conditions for a nearly balance bipartite graph with given minimum degree to be traceable. Secondly, we establish some conditions for a k-connected graph to be Hamilton-connected and traceable for every vertex, respectively.展开更多
The degree distance of a graph G is , where and are the degrees of vertices , and is the distance between them. The Wiener index is defined as . An elegant result (Gutman;Klein, Mihali?,, Plav?i? and Trinajsti?) is kn...The degree distance of a graph G is , where and are the degrees of vertices , and is the distance between them. The Wiener index is defined as . An elegant result (Gutman;Klein, Mihali?,, Plav?i? and Trinajsti?) is known regarding their correlation, that for a tree T with n vertices. In this note, we extend this study for more general graphs that have frequent appearances in the study of these indices. In particular, we develop a formula regarding their correlation, with an error term that is presented with explicit formula as well as sharp bounds for unicyclic graphs and cacti with given parameters.展开更多
Let R be a ring with non-zero identity.The unit graph of R,denoted by G(R),is an undirected graph with all the elements of R as vertices and where distinct vertices x,y are adjacent if and only if x+y is a unit of R.I...Let R be a ring with non-zero identity.The unit graph of R,denoted by G(R),is an undirected graph with all the elements of R as vertices and where distinct vertices x,y are adjacent if and only if x+y is a unit of R.In this paper,we investigate the Wiener index and hyper-Wiener index of G(R)and explicitly determine their values.展开更多
The Merrifield–Simmons indexσis the total number of independent vertex sets(including the empty set)of the graph G.The Wiener index W is the sum of distances in all unordered pairs of vertices of G.We construct some...The Merrifield–Simmons indexσis the total number of independent vertex sets(including the empty set)of the graph G.The Wiener index W is the sum of distances in all unordered pairs of vertices of G.We construct some new graphs satisfyingσ>W and W>σ,respectively.In particular,infinite graphs satisfying W>σare invented with graphs with diameter 2 and infinite ones satisfyingσ>W are discovered with so-called universally diametrical graphs.展开更多
The Wiener index W(G)of a graph G is a distance-based topological index defined as the sum of distances between all pairs of vertices in G.It is shown that forλ=2 there is an infinite family of planar bipartite chemi...The Wiener index W(G)of a graph G is a distance-based topological index defined as the sum of distances between all pairs of vertices in G.It is shown that forλ=2 there is an infinite family of planar bipartite chemical graphs G of girth 4 with the cyclomatic numberλ,but their line graphs are not chemical graphs,and forλ≥2 there are two infinite families of planar nonbipartite graphs G of girth 3 with the cyclomatic numberλ;the three classes of graphs have the property W(G)=W(L(G)),where L(G)is the line graph of G.展开更多
The q-Wiener index of unicyclic graphs are determined in this work. As an example of its applications, an explicit expression of q-Wiener index of caterpillar cycles is presented.
The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vert...The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.12201471)the Special Foundation in Key Fields for Universities of Guangdong Province(Grant No.2022ZDZX1034).
文摘The Wiener index of a graph is defined to be the sum of the distances of all pairs of vertices in the graph.The kth power G^(k) of a graph G is the graph on V(G)and two vertices are adjacent if and only if their distance in G is less or equal to k.In this paper,we computed the Wiener index of the kth power of paths and cycles for any k≥2.
文摘According to the definitions of molecular connectivity and hyper-Wiener index, a novel set of hyper-Wiener indexes （Dn, ^mDn） were defined and named as condensed extended hyper-Wiener index, the potential usefulness of which in QSAR/QSPR is evaluated by its correlation with a number of C3-C8 alkanes as well as by a favorable comparison with models based on molecular connectivity index and overall Wiener index.
文摘The undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) of a finite group <i>Z<sub>n</sub></i> is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if <i>u</i> ≠ <i>v</i> and <img src="Edit_3b1df203-9ff2-4c13-93d1-4bba568eae54.png" width="40" height="20" alt="" /> or <img src="Edit_094c8f88-deb6-4f41-825a-ba91c0306ae8.png" width="40" height="20" alt="" />. The Wiener index <i>W</i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of an undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be sum <img src="Edit_348337df-b9c2-480d-9713-ec299a6fcd4e.png" width="110" height="25" alt="" /> of distances between all unordered pair of vertices in <i>P</i>(<i>Z<sub>n</sub></i>). Similarly, the edge-Wiener index <i>W<sub>e</sub></i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be the sum <img src="Edit_e9b89765-f71e-4865-a0c5-c688710ff0c6.png" width="60" height="25" alt="" /> of distances between all unordered pairs of edges in <i>P</i>(<i>Z<sub>n</sub></i>). In this paper, we concentrate on the wiener index of a power graph <img src="Edit_dff0cd99-eb11-4123-a437-78cbbd8ebf96.png" width="40" height="20" alt="" />, <i>P</i>(<i>Z<sub>pq</sub></i>) and <i>P</i>(<i>Z<sub>p</sub></i>). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph <i>P</i>(<i>Z<sub>n</sub></i>), using <i>m,n</i> and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of <i>Z<sub>n</sub></i>.
基金The National Natural Science Foundation of China-Shandong Joint Fund for Marine Science Research Centers under contract No.U1406403the National Marine Public Welfare Research Project of China under contract No.201405007the Environmental Assessment Projects of State Oceanic Administration People's Republic of China under contract No.DOMEP(MEA)-01-01
文摘From a dataset of macrobenthos obtained from 18 cruises from 2004 to 2013 in the Huanghe（Yellow River）Estuary and its adjacent areas,the composition and characteristics of macrobenthos were analyzed,and the applicability of the Shannon-Wiener Index（H′）,AZTI＇s Marine Biotic Index（AMBI） and multivariate AMBI（MAMBI） for assessing benthic habitat quality was compared.The results showed a total of 203 macrobenthos in the study area.The macrobenthos were dominated by polychaetes,followed by mollusks and crustaceans.The macrobenthic ecological groups were dominated by EGI,EGII and EGIII,which respectively accounted for 31.5%,36.0% and 21.2% of the total.There were significant differences between the evaluation results of the three indices.The ecological quality status（EQS） levels given by the AMBI were greater than those given by the H′ and M-AMBI.The AMBI could not reflect the differences between 11 sites but the H′ and M-AMBI could do.Moreover,the three indices responded well to the variations in salinity（S） and dissolved oxygen（DO） in the waters.The H′ and MAMBI also responded sensitively to the differences in physical parameters,such as water depth and sediment texture.The correlation between M-AMBI and environmental pressure gradient data was the strongest.The MAMBI could effectively distinguish degraded conditions from undegraded but the H′ and AMBI could not.Therefore,the M-AMBI reflected benthic habitat health well in the study areas.However,the objectivity of evaluation results of M-AMBI needs further verification by physical,chemical and biological methods.The thresholds also need further discussion.
基金The NSF(11301093,11501139)of Chinathe NSF(2014A030313640)of Guangdong Provincethe Foundation(Yq2014111)for Distinguished Young Talents in Higher Education of Guangdong Province
文摘In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. Discrete Appl. Math., 2014, 171：35 41）, and give a simple way to compare the Wiener index and the Zagreb indices for trees. Moreover, the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.
基金Supported by the National Natural Science Foun-dation of China(20177018)the Educational Bureau Foundation of Hubei(2003BOO1)
文摘Based on distance matrixW,the novel topological indexW F is defined by the matricesX,W,L,asW F=WXL.WhereX is a row matrix,which expresses the bonding characteristics between adjacent atoms;L is a column matrix expressing the characteristic of vertexes in molecules;W is a distance matrix,which expresses the chemical environment of the atoms in the molecule.the properties,the chemical environments and the interaction of the vertexes in a molecule are taken into account in this definition.Hence the application of the indexW F to multiple bonds and heteroatomcontaining organic systems or inorganic systems is possible.Good correlation coefficients are achieved not only in the prediction of the standard formation enthalpy of methyl halides,titanium halides,but also the retention index of gas chromatography of the hydrocarbons.
文摘The Wiener index W（G） of a graph G is defined as the sum of distances between all pairs of vertices of the graph, Let G＊c, is the set of the complements of bipartite graphs with order n. In this paper, we characterize the graphs with the maximum and second-maximum Wiener indices among all the graphs in G＊c, respectively.
基金Supported by the Natural Science Foundation of China under Grant no(11871077)the NSF of Anhui Province of China under Grant no(1808085MA04)the Natural Science Foundation of Department of Education of Anhui Province of China under Grant no(KJ2017A362)
文摘In this paper, in terms of Wiener index, hyper-Wiener index and Harary index, we first give some sufficient conditions for a nearly balance bipartite graph with given minimum degree to be traceable. Secondly, we establish some conditions for a k-connected graph to be Hamilton-connected and traceable for every vertex, respectively.
文摘The degree distance of a graph G is , where and are the degrees of vertices , and is the distance between them. The Wiener index is defined as . An elegant result (Gutman;Klein, Mihali?,, Plav?i? and Trinajsti?) is known regarding their correlation, that for a tree T with n vertices. In this note, we extend this study for more general graphs that have frequent appearances in the study of these indices. In particular, we develop a formula regarding their correlation, with an error term that is presented with explicit formula as well as sharp bounds for unicyclic graphs and cacti with given parameters.
基金supported in part by a grant from Science and Engineering Research Board,India(Ref.MTR/2017/000830)supported by the National Natural Science Foundation of China(Ref.11961050)the Guangxi Natural Sciences Foundation(Ref.2021GXNSFAA220043).
文摘Let R be a ring with non-zero identity.The unit graph of R,denoted by G(R),is an undirected graph with all the elements of R as vertices and where distinct vertices x,y are adjacent if and only if x+y is a unit of R.In this paper,we investigate the Wiener index and hyper-Wiener index of G(R)and explicitly determine their values.
基金supported by NNSF of China (Grant No. 11671202)supported by National Research Foundation funded by the Korean government (Grant No. 2021R1F1A1050)
文摘The Merrifield–Simmons indexσis the total number of independent vertex sets(including the empty set)of the graph G.The Wiener index W is the sum of distances in all unordered pairs of vertices of G.We construct some new graphs satisfyingσ>W and W>σ,respectively.In particular,infinite graphs satisfying W>σare invented with graphs with diameter 2 and infinite ones satisfyingσ>W are discovered with so-called universally diametrical graphs.
基金This work was supported by the National Natural Science Foundation of China(No.11171273)The authors would like to express their sincere gratitude to the anonymous referees for their comments and remarks,which improved the presentation of this paper.
文摘The Wiener index W(G)of a graph G is a distance-based topological index defined as the sum of distances between all pairs of vertices in G.It is shown that forλ=2 there is an infinite family of planar bipartite chemical graphs G of girth 4 with the cyclomatic numberλ,but their line graphs are not chemical graphs,and forλ≥2 there are two infinite families of planar nonbipartite graphs G of girth 3 with the cyclomatic numberλ;the three classes of graphs have the property W(G)=W(L(G)),where L(G)is the line graph of G.
基金supported by National Natural Science Foundation of China(11126326)NSF of Guandong Province(S2012010010815)Foundation of Wuyi University(201210041650504)
文摘The q-Wiener index of unicyclic graphs are determined in this work. As an example of its applications, an explicit expression of q-Wiener index of caterpillar cycles is presented.
文摘The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.
