作者机构：
[Chen, Shubo] Hunan City Univ, Dept Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.;[Chen, Shubo] Cent South Univ, Coll Math, Changsha 410075, Peoples R China.;[Zhou, Houqing] Shaoyang Univ, Dept Math, Shaoyang 422000, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;[Chen, Shubo] C;Hunan City Univ, Dept Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.;Cent South Univ, Coll Math, Changsha 410075, Peoples R China.
摘要：
The Zagreb indices are topological indices of graphs, which defined as, M-1(G) = Sigma(v is an element of V(G)) (d(v))(2) , M-2(G) = Sigma(uv is an element of E(G)) (d(u)d(v)). In this paper, we determine the upper and lower bounds for the Zagreb indices of unicyclic graphs in terms of their order and girth. In each case, we characterize the extremal graphs.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Coll Math, Yiyang 413000, Hunan, Peoples R China.;Hunan Univ, Coll Math, Changsha 410082, Hunan, Peoples R China.
摘要：
In this paper, we study a new graph invariant named Resistance-Harary index, defined as RH(G) = Sigma({u,v}subset of v(G)) 1/r(G)(u,v) where r(G)(u, v) is the resistance distance between vertices u and v of a connected graph G. We establish that S-n(3) and P-n(3). are the graphs with the maximal and minimal Resistance-Harary index among all unicyclic graphs on n vertices, respectively.
摘要：
The resistance distance r(u, v) between two vertices u, v of a connected graph G is defined as the effective resistance between them in the corresponding electrical network, constructed from G by replacing each edge of G with a unit resistor. The degree resistance distance of G is defined as D-R(G) = Sigma({u, v}subset of) (V(G)) (d(u) + d(upsilon)) r(u, v), where d(u) is the degree of the vertex u. Let G(n,m) be the set of connected graphs with order n and m edges. In this paper, we'll determine the extremal graphs among G(n,m) with the minimum degree resistance distance.
期刊：
Ars Combinatoria,2016年124:183-192 ISSN：0381-7032
通讯作者：
Chen, Shubo
作者机构：
[Chen, Shubo] Hunan City Univ, Dept Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Dept Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
摘要：
Let G be a cactus, which all of blocks of G are either edges or cycles. Denote W(n,r) the set of cactuses of order n and with r cycles. In this paper, we present a unified approach to the extremal cactuses, for Schultz and the modified Schultz indices.
期刊：
Journal of Combinatorial Optimization,2016年31(1):327-332 ISSN：1382-6905
通讯作者：
Chen, Shubo
作者机构：
[Chen, Shubo] Hunan City Univ, Coll Math, Yiyang 413000, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Coll Math, Yiyang 413000, Hunan, Peoples R China.
关键词：
Gutman index;Degree distance;Extremal graph
摘要：
The Gutman index (also known as Schultz index of the second kind) of a graph
$$G$$
is defined as
$$Gut(G)=\sum \nolimits _{u,v\in V(G)}d(u)d(v)d(u, v)$$
. A graph
$$G$$
is called a cactus if each block of
$$G$$
is either an edge or a cycle. Denote by
$$\mathcal {C}(n, k)$$
the set of connected cacti possessing
$$n$$
vertices and
$$k$$
cycles. In this paper, we give the first three smallest Gutman indices among graphs in
$$\mathcal {C}(n, k)$$
, the corresponding extremal graphs are characterized as well.
期刊：
Ars Combinatoria,2016年127:315-327 ISSN：0381-7032
通讯作者：
Chen, Shubo
作者机构：
[Yang, Jianguang; Chen, Shubo] Hunan City Univ, Sch Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Sch Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
摘要：
The resistance distance between two vertices of a connected graph G is defined as the effective resistance between them in the corresponding electrical network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index of K f(G) is the sum of resistance distances between all pairs of vertices of the graph G. In this paper, we'll determine the tricyclic graphs with the smallest and the second smallest Kirchhoff indices.
期刊：
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY,2016年75(1):157-168 ISSN：0340-6253
通讯作者：
Chen, Shubo
作者机构：
[Chen, Qiao; Cai, Xia; Guo, Zhijun; Chen, Shubo] Hunan City Univ, Coll Math, Yiyang 413000, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Coll Math, Yiyang 413000, Hunan, Peoples R China.
摘要：
The resistance distance r(u, v) between two vertices u, v of a connected graph G is defined as the effective resistance between them in the corresponding elec-trical network constructed from G by replacing each edge of G with a unit resis-tor. Let G be a connected graph, the degree resistance distance of G is defined as DR(G) = P-u,v-°°V (G)[d(u) + d(v)]r(u, v), where d(u) is the degree of the vertex u. In this paper, we firstly characterize n-vertex unicyclic graphs with given girth having maximum and second maximum degree resistance distance, then give n-vertex unicyclic graphs with the maximum and second maximum degree resistance distance. of G is defined as u) + d(v)]r(u, v), where d(u) is the degree of the vertex u. In this paper, we firstly characterize n-vertex unicyclic graphs with given girth having maximum and second maximum degree resistance distance, then give n-vertex unicyclic graphs with the maximum and second maximum degree resistance distance.
期刊：
Ars Combinatoria,2015年122:79-88 ISSN：0381-7032
通讯作者：
Chen, Shubo
作者机构：
[Chen, Shubo] Hunan City Univ, Coll Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.;[Liu, Weijun] Cent S Univ, Coll Math, Changsha 410075, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Coll Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
摘要：
For a graph G = (V, E), the Schultz index of G is defined as {equation presented} where dG{u) (or d(u)) is {equation presented} the degree of the vertex u in G, and do{u, v) is the distance between u and v in G. In this paper, we investigate the Schultz index of tricyclic graphs. The n-tricyclic graphs with the minimum Schultz index are determined.
期刊：
Ars Combinatoria,2015年122:379-397 ISSN：0381-7032
通讯作者：
Chen, Shubo
作者机构：
[Chen, Shubo] Hunan City Univ, Sch Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.;[Liu, Weijun; Chen, Shubo] Cent South Univ, Coll Math & Stat, Changsha 410075, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Sch Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
摘要：
For a graph G = (V, E), the modified Schultz index of G is defined as S∗(G) = σ (dG(u).dG(v))dG(u, v) where dG{u) (or d(u)) is {u,u}CV(G) the degree of the vertex u in G, and dG(u, v) is the distance between u and v. The first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. In this paper, we present a unified approach to investigate the modified Schultz index and Zagreb indices of tricyclicgraphs. The tricyclic graph with n vertices having minimum modified Schultz index and maximum Zagreb indicesare determined.
期刊：
Ars Combinatoria,2015年121:19-32 ISSN：0381-7032
通讯作者：
Chen, Shubo
作者机构：
[Chen, Shubo] Hunan City Univ, Coll Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.;Hunan First Normal Univ, Coll Math, Changsha 410205, Hunan, Peoples R China.
通讯机构：
[Chen, Shubo] H;Hunan City Univ, Coll Math & Comp Sci, Yiyang 413000, Hunan, Peoples R China.
摘要：
Let G be a connected graph, the degree resistance distance of G is defined as Dr(G) = d{u) + d(v)]r(u,v), where d(u) (and (<,<}CV(G) d(v)) is the degree of the vertex u (and v), r(u,v) is the resistance distance between vertices u and v. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than 3 in its unique cycle. In this paper, we determine the minimum and maximum degree resistance distance among all fully loaded unicyclic graphs with n vertices, and characterize the extremal graphs.
