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	1. $Z_3$-connectivity for power graphs
	LI Xiangwen
	Mathematics 13 June 2017
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	Abstract:Let $G$ be a connected graph. For an integer $kgeq 2$, $G^k$ isdefined to be a graph obtained from $G$ by adding new edge $uv$where $2leq d(u, v)leq k$. Let $A$ be an Abelian group with$\|A\|geq 3$. In this note, we prove that for any connected graph$G$, $G^l$ is $Z_3$-connected if and only if $\|V(G)\|geq 5$ or$Gcong K_1$, where $lgeq 3$.
	TO cite this article:LI Xiangwen. $Z_3$-connectivity for power graphs[OL].[13 June 2017] http://en.paper.edu.cn/en_releasepaper/content/4736426


	2. The linear programming approach to the harmonic index of a graph
	ZHU Yan,CHANG Ren-ying
	Mathematics 17 April 2017
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	Abstract:The harmonic index $H(G)$ of a graph $G$ is the sum of $rac{2}{d(u)+d(upsilon)}$ over all edges $uupsilon$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we give the minimum value of $H(G)$ for graphs $G$ with given minimum degree $delta(G) geq 3$ and characterize the corresponding extremal graph.
	TO cite this article:ZHU Yan,CHANG Ren-ying. The linear programming approach to the harmonic index of a graph[OL].[17 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4725178

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	3. Unicyclic graphs with the fourth extremal Wiener indices
	YANG Yu-Jun,CAO Yu-Liang,Wang Guang-Fu
	Mathematics 22 March 2017
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	Abstract:Unicyclic graphs with the fourth extremal Wiener indices are characterized. It is shown that among all unicyclic graphs with $ngeq 8$ vertices,$C_5(S_{n-4})$ and $C_2^{u_1,u_2}(S_3,S_{n-4})$ have the fourth minimum Wiener indices, whereas $C^{u_1,u_2}_3(P_{3},P_{n-4})$ has the fourth maximum Wiener index.
	TO cite this article:YANG Yu-Jun,CAO Yu-Liang,Wang Guang-Fu. Unicyclic graphs with the fourth extremal Wiener indices[OL].[22 March 2017] http://en.paper.edu.cn/en_releasepaper/content/4722647


	4. The thickness of the complete bipartite graph $K_{n,n+4}$
	HU Si-Wei,CHEN Yi-Chao
	Mathematics 10 March 2017
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	Abstract:The thickness t(G) of a graph G is the minimum number of planar spanning subgraphsinto which G can be decomposed. Determining the thickness of a graphis NP-hard, thus it is very difficult to obtain the exact number of thickness forarbitrary graphs. Compared with other classical topological invariant, the results about thickness are few. For the thickness of complete bipartite graph $K_{m,n}$, Beineke, Harary and Moon obtained the following formula in 1964 ,$$t(K_{m,n})=leftlceilrac{mn}{2(m+n-2)} ight ceil,$$ except possibly when $m$ and $n$ are both odd and there exists an integer $k$ satisfying $n=leftlfloorrac{2k(m-2)}{(m-2k)} ight floor$.According to the above formula, the thickness of the complete bipartite graph is not completely determined. In 1980, two famous graph theorist Gross and Harary posed the following problem in the paper 《 Some problems in topological graph theory》： Find the thickness of $K_{m,n}$ for all $m,n?$ In this paper, we obtain the thickness for the complete bipartite graph $K_{n,n+4} .$
	TO cite this article:HU Si-Wei,CHEN Yi-Chao. The thickness of the complete bipartite graph $K_{n,n+4}$[OL].[10 March 2017] http://en.paper.edu.cn/en_releasepaper/content/4721674


	5. On distance signless Laplacian spectral radius of graphs
	KE Xiao-xia, ZHOU Bo
	Mathematics 21 May 2016
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	Abstract:We determine the unique graphs with minimum distance signless Laplacian spectral radius among bipartite graphs with given matching number, among graphs with given independent number and number of pendent vertices, among graphs with given number of odd vertices. We also determine the unique tree with second minimum distance signless Laplacian spectral radius among trees with all vertices being odd and the unique graph with minimum distance signless Laplacian spectral radius among unicyclic graphs with all vertices being odd.
	TO cite this article:KE Xiao-xia, ZHOU Bo. On distance signless Laplacian spectral radius of graphs[OL].[21 May 2016] http://en.paper.edu.cn/en_releasepaper/content/4692306


	6. Nodal domain count and vertex bipartiteness
	LUO Zuo-juan, ZHOU Bo
	Mathematics 28 November 2015
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	Abstract:We establish a novel relation between the nodal domain count and the vertex bipartiteness of a graph, give upper and/or lower bound for the nodal domain count of a graph in terms ofthe independent number, and the diameter, and the chromatic number, and characterize the (connected) graphs $G$ withnodal domain count $4$.
	TO cite this article:LUO Zuo-juan, ZHOU Bo. Nodal domain count and vertex bipartiteness[OL].[28 November 2015] http://en.paper.edu.cn/en_releasepaper/content/4666784


	7. Extremal Graphs with Maximum Edge-Neighbor-Connectivity
	BAI Yan-Ru, ZHANG Zhao, LIU Qing-Hai
	Mathematics 26 August 2015
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	Abstract:An edge is subverted if the two ends of the edge aredeleted from the graph. The edge-neighbor-connectivity$lambda_{NB}(G)$ is the minimum number of edges the subversion ofwhich results in an empty, or trivial, or disconnected graph. It isknown that $lambda_{NB}(G)leqlfloor n/2 floor$, where$n=\|V(G)\|$. In this paper, we characterize all extremal graphs whoseedge-neighbor-connectivity reaches this upper bound: for even $n$,an extremal graph can only be the complete graph $K_n$ or thecomplete bipartite graph $K_{rac{n}{2},rac{n}{2}}$; for odd $n$,an extremal graph can only be the 5-cycle $C_5$, or $K_n-M_0$ (thecomplete graph with a matching $M_0$ removed, where $M_0$ is anarbitrary matching of $K_n$ containing $i$ edges for $iin{0,1,ldots,lfloor n/2 floor}$), or a graph $G$ spanned by a$K_{lfloor rac{n}{2} floor,lceil rac{n}{2} ceil}$ such thatthe $lfloor n/2 floor$-part is independent in $G$ and the subgraphinduced by the $lceil n/2 ceil$-part has matching number at mostone.
	TO cite this article:BAI Yan-Ru, ZHANG Zhao, LIU Qing-Hai. Extremal Graphs with Maximum Edge-Neighbor-Connectivity[OL].[26 August 2015] http://en.paper.edu.cn/en_releasepaper/content/4653169


	8. Diameter variation of directed cycles and directed tori
	MA Xiao-Yan, Huang Xiao-Hui, ZHANG Zhao
	Mathematics 26 August 2015
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	Abstract:In this paper, we study two parameters concerningwith the diameter variation under the addition or deletion of arcsin a digraph $G$: $D^{-0}(G)$ is the maximum number of arcs theaddition of which dose not change the diameter of $G$; $D^{+k}(G)$is the minimum number of arcs the deletion of which increases thediameter of $G$ by at least $k$. We give a formula for $D^{-0}(G)$if $G$ is vertex transitive and every vertex has a unique vertexwhich is farthest from it. As consequences, the values of $D^{-0}$for the directed cycle and the directed torus can be determined. Thevalues of $D^{+k}$ for these two digraphs are also determined.
	TO cite this article:MA Xiao-Yan, Huang Xiao-Hui, ZHANG Zhao. Diameter variation of directed cycles and directed tori[OL].[26 August 2015] http://en.paper.edu.cn/en_releasepaper/content/4653178


	9. Rainbow matchings and minimum color degree
	LI Hua-Long, WANG Guang-Hui, YAN Gui-Ying
	Mathematics 20 December 2013
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	Abstract:Let $G$ be an edge-colored graph. A rainbowmatching of $G$ is a matching in which no two edges have thesame color. Let $hat{delta}$ denote the minimum color degree of $G$, i.e. the smallest number of distinct colors on the edges incident with a vertex over all vertices. We show that if $\|V(G)\|geq 4hat{delta}-5$ when $hat{delta}geq 4$,then $G$ has a rainbow matching of size $hat{delta}$, which improves the previous result.
	TO cite this article:LI Hua-Long, WANG Guang-Hui, YAN Gui-Ying. Rainbow matchings and minimum color degree[OL].[20 December 2013] http://en.paper.edu.cn/en_releasepaper/content/4576838


	10. On the harmonic index of acyclic conjugated molecular graphs
	ZHU Yan, CHANG Ren-ying
	Mathematics 18 December 2013
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	Abstract:The harmonic index $H(G)$ of a graph $G$ is defined as the sum ofweights $rac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$denotes the degree of a vertex $u$ in $G$. In this paper, we firstpresent a sharp lower bound on the harmonic index of acyclicconjugated molecular graphs (trees with a perfect matching). A sharplower bound on the harmonic index of acyclic graphs is also given interms of the order and given size of matching.
	TO cite this article:ZHU Yan, CHANG Ren-ying. On the harmonic index of acyclic conjugated molecular graphs[OL].[18 December 2013] http://en.paper.edu.cn/en_releasepaper/content/4575821

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