A strong product graph is denoted by G_(1)■G_(2),where G_(1) and G_(2) are called its factor graphs.This paper gives the range of the minimum strong radius of the strong product graph.And using the relationship betwe...A strong product graph is denoted by G_(1)■G_(2),where G_(1) and G_(2) are called its factor graphs.This paper gives the range of the minimum strong radius of the strong product graph.And using the relationship between the cartesian product graph G_(1)■G_(2) and the strong product graph G_(1)■G_(2),another different upper bound of the minimum strong radius of the strong product graph is given.展开更多
An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The ...An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The L（2, 1）-labeling number λ（G） of G is the smallest number k such that G has an L（2, 1）-labeling with max{f（v） ： v ∈ V（G）} = k. We study the L（3, 2, 1）-labeling which is a generalization of the L（2, 1）-labeling on the graph formed by the （Cartesian） product and composition of 3 graphs and derive the upper bounds of λ3（G） of the graph.展开更多
The acquisition of valuable design knowledge from massive fragmentary data is challenging for designers in conceptual product design.This study proposes a novel method for acquiring design knowledge by combining deep ...The acquisition of valuable design knowledge from massive fragmentary data is challenging for designers in conceptual product design.This study proposes a novel method for acquiring design knowledge by combining deep learning with knowledge graph.Specifically,the design knowledge acquisition method utilises the knowledge extraction model to extract design-related entities and relations from fragmentary data,and further constructs the knowledge graph to support design knowledge acquisition for conceptual product design.Moreover,the knowledge extraction model introduces ALBERT to solve memory limitation and communication overhead in the entity extraction module,and uses multi-granularity information to overcome segmentation errors and polysemy ambiguity in the relation extraction module.Experimental comparison verified the effectiveness and accuracy of the proposed knowledge extraction model.The case study demonstrated the feasibility of the knowledge graph construction with real fragmentary porcelain data and showed the capability to provide designers with interconnected and visualised design knowledge.展开更多
Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that an...Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.展开更多
In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with...In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with stars （presented by Marian Klesc）. In addition, we also give an up to date description of Cartesian products of 5-vertex graphs with stars, whose crossing numbers are known.展开更多
L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the...L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the join graph on two fans is?λ(G) ≤ 2Δ+3.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11551002)Natural Science Foundation of Qinghai Province(Grant No.2019-ZJ-7093)。
文摘A strong product graph is denoted by G_(1)■G_(2),where G_(1) and G_(2) are called its factor graphs.This paper gives the range of the minimum strong radius of the strong product graph.And using the relationship between the cartesian product graph G_(1)■G_(2) and the strong product graph G_(1)■G_(2),another different upper bound of the minimum strong radius of the strong product graph is given.
文摘An L（2, 1）-labeling of a graph G is a function f from the vertex set V（G） to the set of all nonnegative integers such that ｜f（x） - f（y）｜ 〉 2 if d（x, y） = 1 and ｜f（x）-f（y）｜ ≥ 1 ifd（x, y） = 2. The L（2, 1）-labeling number λ（G） of G is the smallest number k such that G has an L（2, 1）-labeling with max{f（v） ： v ∈ V（G）} = k. We study the L（3, 2, 1）-labeling which is a generalization of the L（2, 1）-labeling on the graph formed by the （Cartesian） product and composition of 3 graphs and derive the upper bounds of λ3（G） of the graph.
基金This research is supported by the Chinese Special Projects of the National Key Research and Development Plan(2019YFB1405702).
文摘The acquisition of valuable design knowledge from massive fragmentary data is challenging for designers in conceptual product design.This study proposes a novel method for acquiring design knowledge by combining deep learning with knowledge graph.Specifically,the design knowledge acquisition method utilises the knowledge extraction model to extract design-related entities and relations from fragmentary data,and further constructs the knowledge graph to support design knowledge acquisition for conceptual product design.Moreover,the knowledge extraction model introduces ALBERT to solve memory limitation and communication overhead in the entity extraction module,and uses multi-granularity information to overcome segmentation errors and polysemy ambiguity in the relation extraction module.Experimental comparison verified the effectiveness and accuracy of the proposed knowledge extraction model.The case study demonstrated the feasibility of the knowledge graph construction with real fragmentary porcelain data and showed the capability to provide designers with interconnected and visualised design knowledge.
基金This work was supported by the National Natural Science Foundation of China(Grant No. 10001005) and by RFDP of China.
文摘Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.
基金the National Natural Science Foundation of China （No. 10771062） the Fund for New Century Excellent Talents in University （No. NCET-07-0276）.
文摘In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with stars （presented by Marian Klesc）. In addition, we also give an up to date description of Cartesian products of 5-vertex graphs with stars, whose crossing numbers are known.
文摘L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is?λ(G) ≤ Δ+3 , L(2,1)-labeling number of the join graph on two fans is?λ(G) ≤ 2Δ+3.
