With the advancement of Vehicle-to-Everything(V2X)technology,efficient resource allocation in dynamic vehicular networks has become a critical challenge for achieving optimal performance.Existing methods suffer from h...With the advancement of Vehicle-to-Everything(V2X)technology,efficient resource allocation in dynamic vehicular networks has become a critical challenge for achieving optimal performance.Existing methods suffer from high computational complexity and decision latency under high-density traffic and heterogeneous network conditions.To address these challenges,this study presents an innovative framework that combines Graph Neural Networks(GNNs)with a Double Deep Q-Network(DDQN),utilizing dynamic graph structures and reinforcement learning.An adaptive neighbor sampling mechanism is introduced to dynamically select the most relevant neighbors based on interference levels and network topology,thereby improving decision accuracy and efficiency.Meanwhile,the framework models communication links as nodes and interference relationships as edges,effectively capturing the direct impact of interference on resource allocation while reducing computational complexity and preserving critical interaction information.Employing an aggregation mechanism based on the Graph Attention Network(GAT),it dynamically adjusts the neighbor sampling scope and performs attention-weighted aggregation based on node importance,ensuring more efficient and adaptive resource management.This design ensures reliable Vehicle-to-Vehicle(V2V)communication while maintaining high Vehicle-to-Infrastructure(V2I)throughput.The framework retains the global feature learning capabilities of GNNs and supports distributed network deployment,allowing vehicles to extract low-dimensional graph embeddings from local observations for real-time resource decisions.Experimental results demonstrate that the proposed method significantly reduces computational overhead,mitigates latency,and improves resource utilization efficiency in vehicular networks under complex traffic scenarios.This research not only provides a novel solution to resource allocation challenges in V2X networks but also advances the application of DDQN in intelligent transportation systems,offering substantial theoretical significance and practical value.展开更多
With the development of the Semantic Web,the number of ontologies grows exponentially and the semantic relationships between ontologies become more and more complex,understanding the true semantics of specific terms o...With the development of the Semantic Web,the number of ontologies grows exponentially and the semantic relationships between ontologies become more and more complex,understanding the true semantics of specific terms or concepts in an ontology is crucial for the matching task.At present,the main challenges facing ontology matching tasks based on representation learning methods are how to improve the embedding quality of ontology knowledge and how to integrate multiple features of ontology efficiently.Therefore,we propose an Ontology Matching Method Based on the Gated Graph Attention Model(OM-GGAT).Firstly,the semantic knowledge related to concepts in the ontology is encoded into vectors using the OWL2Vec^(*)method,and the relevant path information from the root node to the concept is embedded to understand better the true meaning of the concept itself and the relationship between concepts.Secondly,the ontology is transformed into the corresponding graph structure according to the semantic relation.Then,when extracting the features of the ontology graph nodes,different attention weights are assigned to each adjacent node of the central concept with the help of the attention mechanism idea.Finally,gated networks are designed to further fuse semantic and structural embedding representations efficiently.To verify the effectiveness of the proposed method,comparative experiments on matching tasks were carried out on public datasets.The results show that the OM-GGAT model can effectively improve the efficiency of ontology matching.展开更多
High-entropy alloy(HEA)offer tunable composition and surface structures,enabling the creation of novel active sites that enhance catalytic performance in renewable energy application.However,the inherent surface compl...High-entropy alloy(HEA)offer tunable composition and surface structures,enabling the creation of novel active sites that enhance catalytic performance in renewable energy application.However,the inherent surface complexity and tendency for elemental segregation,which results in discrepancies between bulk and surface compositions,pose challenges for direct investigation via density functional theory.To address this,Monte Carlo simulations combined with molecular dynamics were employed to model surface segregation across a broad range of elements,including Cu,Ag,Au,Pt,Pd,and Al.The analysis revealed a trend in surface segregation propensity following the order Ag>Au>Al>Cu>Pd>Pt.To capture the correlation between surface site characteristics and the free energy of multi-dentate CO_(2)reduction intermediates,a graph neural network was designed,where adsorbates were transformed into pseudo-atoms at their centers of mass.This model achieved mean absolute errors of 0.08–0.15 eV for the free energies of C_(2)intermediates,enabling precise site activity quantification.Results indicated that increasing the concentration of Cu,Ag,and Al significantly boosts activity for CO and C_(2)formation,whereas Au,Pd,and Pt exhibit negative effects.By screening stable composition space,promising HEA bulk compositions for CO,HCOOH,and C_(2)products were predicted,offering superior catalytic activity compared to pure Cu catalysts.展开更多
In this paper,we prove that there does not exist an r-UPC[2]-graph for each r≥5 and there does not exist an r-UPC[C_t^2]-graph for each r≥3,where t is the number of bridges in a graph and C_t^2 is the number of comb...In this paper,we prove that there does not exist an r-UPC[2]-graph for each r≥5 and there does not exist an r-UPC[C_t^2]-graph for each r≥3,where t is the number of bridges in a graph and C_t^2 is the number of combinations of t bridges taken 2 at a time.展开更多
The performance of the graph-based scheduling for device-to-device communications overlaying cellular networks is studied. The graph-based scheduling consists of two stages, the frequency assignment stage and the time...The performance of the graph-based scheduling for device-to-device communications overlaying cellular networks is studied. The graph-based scheduling consists of two stages, the frequency assignment stage and the time slot scheduling stage. For such scheduling, a theoretical method to analyze the average spectrum efficiency of the D2D subsystem is proposed. The method consists of three steps. First, the frequency assignment stage is analyzed and the approximate formula of the average number of the D2D links which are assigned the same frequency is derived. Secondly, the time slot scheduling stage is analyzed and the approximate formula of the average probability of a D2D link being scheduled in a time slot is derived. Thirdly, the average spectrum efficiency of the D2D subsystem is analyzed and the corresponding approximate formula is derived. Analysis results show that the average spectrum efficiency of the D2D subsystem is approximately inversely linearly proportional to the second- order origin moment of the normalized broadcast radius of D2D links. Simulation results show that the proposed method can correctly predict the average spectrum efficiency of the D2D subsystem.展开更多
A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic the...A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.展开更多
In this paper,we jointly design the power control and position dispatch for Multi-Unmanned Aerial Vehicle(UAV)-enabled communication in Device-to-Device(D2D)networks.Our objective is to maximize the total transmission...In this paper,we jointly design the power control and position dispatch for Multi-Unmanned Aerial Vehicle(UAV)-enabled communication in Device-to-Device(D2D)networks.Our objective is to maximize the total transmission rate of Downlink Users(DUs).Meanwhile,the Quality of Service(QoS)of all D2D users must be satisfied.We comprehensively considered the interference among D2D communications and downlink transmissions.The original problem is strongly non-convex,which requires high computational complexity for traditional optimization methods.And to make matters worse,the results are not necessarily globally optimal.In this paper,we propose a novel Graph Neural Networks(GNN)based approach that can map the considered system into a specific graph structure and achieve the optimal solution in a low complexity manner.Particularly,we first construct a GNN-based model for the proposed network,in which the transmission links and interference links are formulated as vertexes and edges,respectively.Then,by taking the channel state information and the coordinates of ground users as the inputs,as well as the location of UAVs and the transmission power of all transmitters as outputs,we obtain the mapping from inputs to outputs through training the parameters of GNN.Simulation results verified that the way to maximize the total transmission rate of DUs can be extracted effectively via the training on samples.Moreover,it also shows that the performance of proposed GNN-based method is better than that of traditional means.展开更多
A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph. In this paper, we show some structural properties that a 2-walk （a, b）-linear graph holds. According to these properties, we can e...A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph. In this paper, we show some structural properties that a 2-walk （a, b）-linear graph holds. According to these properties, we can estimate and characterize more 2-walk linear graphs that have exactly two main eigenvalues.展开更多
The 2-step domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or at distance two from some vertex of D. In the present paper, by using a labeling method,...The 2-step domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or at distance two from some vertex of D. In the present paper, by using a labeling method, we provide an O（m） time algorithm to solve the 2-step domination problem on block graphs, a superclass of trees.展开更多
A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set ...A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].展开更多
Let G = (V;E) be a simple connected graph. The Wiener index is the sum of distances between all pairs of vertices of a connected graph. The Schultz topological index is equal to and the Modified Schultz topological in...Let G = (V;E) be a simple connected graph. The Wiener index is the sum of distances between all pairs of vertices of a connected graph. The Schultz topological index is equal to and the Modified Schultz topological index is . In this paper, the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs J<sub>2,m</sub> for all integer number m ≥ 3 are calculated.展开更多
In this paper we show that the face-width of any embedding of a Halin graph（a type of planar graph） in the torus is one, and give a formula for determining the number of all nonequivalent embeddings of a Halin graph...In this paper we show that the face-width of any embedding of a Halin graph（a type of planar graph） in the torus is one, and give a formula for determining the number of all nonequivalent embeddings of a Halin graph in the torus.展开更多
In this paper we prove that the join of two path graphs, two cycle graphs, Ladder graph and the tensor product are H2-cordial labeling. Further we prove that the join of two wheel graphs Wn and Wm, (mod 4) admits a H-...In this paper we prove that the join of two path graphs, two cycle graphs, Ladder graph and the tensor product are H2-cordial labeling. Further we prove that the join of two wheel graphs Wn and Wm, (mod 4) admits a H-cordial labeling.展开更多
A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken ove...A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.展开更多
In this study, the SK, SK<sub>1</sub> and SK<sub>2</sub> indices are defined on weighted graphs. Then, the SK, SK<sub>1</sub> and SK<sub>2</sub> indices are defined on i...In this study, the SK, SK<sub>1</sub> and SK<sub>2</sub> indices are defined on weighted graphs. Then, the SK, SK<sub>1</sub> and SK<sub>2</sub> indices are defined on interval weighted graphs. Their behaviors are investigated under some graph operations by using these definitions.展开更多
A list assignment of a graph G is a function L:V(G)∪E(G)→2^(N).A graph G is L-(2,1)-Total labeling if there exists a function c such that c(x)∈L(x)for all x∈V(G)∪E(G),|c(u)-c(v)|≥1 if uv∈E(G),|c(e_(1))-c(e_(2))...A list assignment of a graph G is a function L:V(G)∪E(G)→2^(N).A graph G is L-(2,1)-Total labeling if there exists a function c such that c(x)∈L(x)for all x∈V(G)∪E(G),|c(u)-c(v)|≥1 if uv∈E(G),|c(e_(1))-c(e_(2))|≥1 if the edges e_(1)and e_(2)are adjacent,and|c(u)-c(e)|≥2 if the vertex u is incident to the edge e.A graph G is k-(2,1)-Total choosable if G is L-(2,1)-Total labeling for every list assignment L provided that|L(x)|=k,x∈V(G)∪E(G).The(2,1)-Total choice number of G,denoted by C_(2,1)^T(G),is the minimum k such that G is k-(2,1)-Total choosable.In this paper,we prove that if G is a planar graph with△(G)≥11,then C_(2,1)^T(G)≤△+4.展开更多
A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this pape...A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.展开更多
基金Project ZR2023MF111 supported by Shandong Provincial Natural Science Foundation。
文摘With the advancement of Vehicle-to-Everything(V2X)technology,efficient resource allocation in dynamic vehicular networks has become a critical challenge for achieving optimal performance.Existing methods suffer from high computational complexity and decision latency under high-density traffic and heterogeneous network conditions.To address these challenges,this study presents an innovative framework that combines Graph Neural Networks(GNNs)with a Double Deep Q-Network(DDQN),utilizing dynamic graph structures and reinforcement learning.An adaptive neighbor sampling mechanism is introduced to dynamically select the most relevant neighbors based on interference levels and network topology,thereby improving decision accuracy and efficiency.Meanwhile,the framework models communication links as nodes and interference relationships as edges,effectively capturing the direct impact of interference on resource allocation while reducing computational complexity and preserving critical interaction information.Employing an aggregation mechanism based on the Graph Attention Network(GAT),it dynamically adjusts the neighbor sampling scope and performs attention-weighted aggregation based on node importance,ensuring more efficient and adaptive resource management.This design ensures reliable Vehicle-to-Vehicle(V2V)communication while maintaining high Vehicle-to-Infrastructure(V2I)throughput.The framework retains the global feature learning capabilities of GNNs and supports distributed network deployment,allowing vehicles to extract low-dimensional graph embeddings from local observations for real-time resource decisions.Experimental results demonstrate that the proposed method significantly reduces computational overhead,mitigates latency,and improves resource utilization efficiency in vehicular networks under complex traffic scenarios.This research not only provides a novel solution to resource allocation challenges in V2X networks but also advances the application of DDQN in intelligent transportation systems,offering substantial theoretical significance and practical value.
基金supported by the National Natural Science Foundation of China(grant numbers 62267005 and 42365008)the Guangxi Collaborative Innovation Center of Multi-Source Information Integration and Intelligent Processing.
文摘With the development of the Semantic Web,the number of ontologies grows exponentially and the semantic relationships between ontologies become more and more complex,understanding the true semantics of specific terms or concepts in an ontology is crucial for the matching task.At present,the main challenges facing ontology matching tasks based on representation learning methods are how to improve the embedding quality of ontology knowledge and how to integrate multiple features of ontology efficiently.Therefore,we propose an Ontology Matching Method Based on the Gated Graph Attention Model(OM-GGAT).Firstly,the semantic knowledge related to concepts in the ontology is encoded into vectors using the OWL2Vec^(*)method,and the relevant path information from the root node to the concept is embedded to understand better the true meaning of the concept itself and the relationship between concepts.Secondly,the ontology is transformed into the corresponding graph structure according to the semantic relation.Then,when extracting the features of the ontology graph nodes,different attention weights are assigned to each adjacent node of the central concept with the help of the attention mechanism idea.Finally,gated networks are designed to further fuse semantic and structural embedding representations efficiently.To verify the effectiveness of the proposed method,comparative experiments on matching tasks were carried out on public datasets.The results show that the OM-GGAT model can effectively improve the efficiency of ontology matching.
文摘High-entropy alloy(HEA)offer tunable composition and surface structures,enabling the creation of novel active sites that enhance catalytic performance in renewable energy application.However,the inherent surface complexity and tendency for elemental segregation,which results in discrepancies between bulk and surface compositions,pose challenges for direct investigation via density functional theory.To address this,Monte Carlo simulations combined with molecular dynamics were employed to model surface segregation across a broad range of elements,including Cu,Ag,Au,Pt,Pd,and Al.The analysis revealed a trend in surface segregation propensity following the order Ag>Au>Al>Cu>Pd>Pt.To capture the correlation between surface site characteristics and the free energy of multi-dentate CO_(2)reduction intermediates,a graph neural network was designed,where adsorbates were transformed into pseudo-atoms at their centers of mass.This model achieved mean absolute errors of 0.08–0.15 eV for the free energies of C_(2)intermediates,enabling precise site activity quantification.Results indicated that increasing the concentration of Cu,Ag,and Al significantly boosts activity for CO and C_(2)formation,whereas Au,Pd,and Pt exhibit negative effects.By screening stable composition space,promising HEA bulk compositions for CO,HCOOH,and C_(2)products were predicted,offering superior catalytic activity compared to pure Cu catalysts.
文摘In this paper,we prove that there does not exist an r-UPC[2]-graph for each r≥5 and there does not exist an r-UPC[C_t^2]-graph for each r≥3,where t is the number of bridges in a graph and C_t^2 is the number of combinations of t bridges taken 2 at a time.
基金The National Natural Science Foundation of China(No.61571111)the National High Technology Research and Development Program of China(863 Program)(No.2014AA01A703,2015AA01A706)the Fundamental Research Funds for the Central Universities of China(No.2242016K40098)
文摘The performance of the graph-based scheduling for device-to-device communications overlaying cellular networks is studied. The graph-based scheduling consists of two stages, the frequency assignment stage and the time slot scheduling stage. For such scheduling, a theoretical method to analyze the average spectrum efficiency of the D2D subsystem is proposed. The method consists of three steps. First, the frequency assignment stage is analyzed and the approximate formula of the average number of the D2D links which are assigned the same frequency is derived. Secondly, the time slot scheduling stage is analyzed and the approximate formula of the average probability of a D2D link being scheduled in a time slot is derived. Thirdly, the average spectrum efficiency of the D2D subsystem is analyzed and the corresponding approximate formula is derived. Analysis results show that the average spectrum efficiency of the D2D subsystem is approximately inversely linearly proportional to the second- order origin moment of the normalized broadcast radius of D2D links. Simulation results show that the proposed method can correctly predict the average spectrum efficiency of the D2D subsystem.
基金Supported by the National Natural Science Foundation of China (10671081)
文摘A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.
基金supported in part by the National Natural Science Foundation of China(61901231)in part by the National Natural Science Foundation of China(61971238)+3 种基金in part by the Natural Science Foundation of Jiangsu Province of China(BK20180757)in part by the open project of the Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space,Ministry of Industry and Information Technology(KF20202102)in part by the China Postdoctoral Science Foundation under Grant(2020M671480)in part by the Jiangsu Planned Projects for Postdoctoral Research Funds(2020z295).
文摘In this paper,we jointly design the power control and position dispatch for Multi-Unmanned Aerial Vehicle(UAV)-enabled communication in Device-to-Device(D2D)networks.Our objective is to maximize the total transmission rate of Downlink Users(DUs).Meanwhile,the Quality of Service(QoS)of all D2D users must be satisfied.We comprehensively considered the interference among D2D communications and downlink transmissions.The original problem is strongly non-convex,which requires high computational complexity for traditional optimization methods.And to make matters worse,the results are not necessarily globally optimal.In this paper,we propose a novel Graph Neural Networks(GNN)based approach that can map the considered system into a specific graph structure and achieve the optimal solution in a low complexity manner.Particularly,we first construct a GNN-based model for the proposed network,in which the transmission links and interference links are formulated as vertexes and edges,respectively.Then,by taking the channel state information and the coordinates of ground users as the inputs,as well as the location of UAVs and the transmission power of all transmitters as outputs,we obtain the mapping from inputs to outputs through training the parameters of GNN.Simulation results verified that the way to maximize the total transmission rate of DUs can be extracted effectively via the training on samples.Moreover,it also shows that the performance of proposed GNN-based method is better than that of traditional means.
基金Supported by the National Natural Science Foundation of China(11171129)
文摘A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph. In this paper, we show some structural properties that a 2-walk （a, b）-linear graph holds. According to these properties, we can estimate and characterize more 2-walk linear graphs that have exactly two main eigenvalues.
基金Supported by the National Natural Science Foundation of China(Grant No.11271365)the Domestic Senior Visiting Scholar Program in Higher Occupation Colleges in Jiangsu Province(Grant No.2014FX075)
文摘The 2-step domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or at distance two from some vertex of D. In the present paper, by using a labeling method, we provide an O（m） time algorithm to solve the 2-step domination problem on block graphs, a superclass of trees.
文摘A set D of vertices of a graph G = (V, E) is called k-dominating if every vertex v ∈V-D is adjacent to some k vertices of D. The k-domination number of a graph G, γk (G), is the order of a smallest k-dominating set of G. In this paper we calculate the k-domination number (for k = 2) of the product of two paths Pm × Pn for m = 1, 2, 3, 4, 5 and arbitrary n. These results were shown an error in the paper [1].
文摘Let G = (V;E) be a simple connected graph. The Wiener index is the sum of distances between all pairs of vertices of a connected graph. The Schultz topological index is equal to and the Modified Schultz topological index is . In this paper, the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs J<sub>2,m</sub> for all integer number m ≥ 3 are calculated.
基金Supported by the NNSF of China(10671073)Supported by the NSF of Jiangsu’s Universities( 07KJB110090)
文摘In this paper we show that the face-width of any embedding of a Halin graph（a type of planar graph） in the torus is one, and give a formula for determining the number of all nonequivalent embeddings of a Halin graph in the torus.
文摘In this paper we prove that the join of two path graphs, two cycle graphs, Ladder graph and the tensor product are H2-cordial labeling. Further we prove that the join of two wheel graphs Wn and Wm, (mod 4) admits a H-cordial labeling.
文摘A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.
文摘In this study, the SK, SK<sub>1</sub> and SK<sub>2</sub> indices are defined on weighted graphs. Then, the SK, SK<sub>1</sub> and SK<sub>2</sub> indices are defined on interval weighted graphs. Their behaviors are investigated under some graph operations by using these definitions.
基金Supported by the National Natural Science Foundation of China(Grant No.12071265)the Natural Science Foundation of Shandong Province(Grant No.ZR2019MA032)。
文摘A list assignment of a graph G is a function L:V(G)∪E(G)→2^(N).A graph G is L-(2,1)-Total labeling if there exists a function c such that c(x)∈L(x)for all x∈V(G)∪E(G),|c(u)-c(v)|≥1 if uv∈E(G),|c(e_(1))-c(e_(2))|≥1 if the edges e_(1)and e_(2)are adjacent,and|c(u)-c(e)|≥2 if the vertex u is incident to the edge e.A graph G is k-(2,1)-Total choosable if G is L-(2,1)-Total labeling for every list assignment L provided that|L(x)|=k,x∈V(G)∪E(G).The(2,1)-Total choice number of G,denoted by C_(2,1)^T(G),is the minimum k such that G is k-(2,1)-Total choosable.In this paper,we prove that if G is a planar graph with△(G)≥11,then C_(2,1)^T(G)≤△+4.
文摘A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.
