In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥ 1, t ≥ 1 , let n = 2t ＋ 3,2t ＋ 4 , which proved W. U K （1） p,t U K（2） is graceful; for p≥1, t≥1 ...In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥ 1, t ≥ 1 , let n = 2t ＋ 3,2t ＋ 4 , which proved W. U K （1） p,t U K（2） is graceful; for p≥1, t≥1 ,let n=2t＋3,2t＋4, then Wn,2n＋1 U K（1）p,t U K（2） p,t is graceful and for m ≥ 1, r ≥ 1 , let n = 2m ＋ 5, Wn,2n＋1 U （C3 v Km） U St（r） is graceful.展开更多
Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P...Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P^((3))_(1)∨K_((2))^(2n))were presented.For natural number n∈N,n≥1,the two graphs are all graceful graphs,where P^((1))_(2n),P^((2))_(2n)are even-vertices path,P_(2n+1)is odd-vertices path,K_(2n),K^((1))_(2n),K^((2))_(2n)are the complement of graph K_(2 n),G_(1)∨G_(2)is the join graph of G_(1)and G_(2).展开更多
Understanding individual decisions in a world where communications and information move instantly via cell phones and the internet,contributes to the development and implementation of policies aimed at stopping or ame...Understanding individual decisions in a world where communications and information move instantly via cell phones and the internet,contributes to the development and implementation of policies aimed at stopping or ameliorating the spread of diseases.In this manuscript,the role of official social network perturbations generated by public health officials to slow down or stop a disease outbreak are studied over distinct classes of static social networks.The dynamics are stochastic in nature with individuals(nodes)being assigned fixed levels of education or wealth.Nodes may change their epidemiological status from susceptible,to infected and to recovered.Most importantly,it is assumed that when the prevalence reaches a pre-determined threshold level,P*,information,called awareness in our framework,starts to spread,a process triggered by public health authorities.Information is assumed to spread over the same static network and whether or not one becomes a temporary informer,is a function of his/her level of education or wealth and epidemiological status.Stochastic simulations show that threshold selection Pand the value of the average basic reproduction number impact the final epidemic size differentially.For the ErdÖos-Rényi and Small-world networks,an optimal choice for Pthat minimize the final epidemic size can be identified under some conditions while for Scalefree networks this is not case.展开更多
基金Supported by the Natural Science Foundation of Beijing(1102015)University Scientific Research Project of Hebei Province(Z2014032)the Fundamental Research Funds for the Central Universities(HKXJZD201402,2011B019,3142013025,3142014127)
文摘In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥ 1, t ≥ 1 , let n = 2t ＋ 3,2t ＋ 4 , which proved W. U K （1） p,t U K（2） is graceful; for p≥1, t≥1 ,let n=2t＋3,2t＋4, then Wn,2n＋1 U K（1）p,t U K（2） p,t is graceful and for m ≥ 1, r ≥ 1 , let n = 2m ＋ 5, Wn,2n＋1 U （C3 v Km） U St（r） is graceful.
基金the National Natural Science Foundation of China(11702094)the Fundamental Research Funds for the Central University(3142015045)。
文摘Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P^((3))_(1)∨K_((2))^(2n))were presented.For natural number n∈N,n≥1,the two graphs are all graceful graphs,where P^((1))_(2n),P^((2))_(2n)are even-vertices path,P_(2n+1)is odd-vertices path,K_(2n),K^((1))_(2n),K^((2))_(2n)are the complement of graph K_(2 n),G_(1)∨G_(2)is the join graph of G_(1)and G_(2).
基金This work was supported by the grant from the National Security Agency(NSAGrantH98230-J8-1-0005)National Science Foundation(NSF Grant 1716802)James S.McDonnell Foundation(220020472)。
文摘Understanding individual decisions in a world where communications and information move instantly via cell phones and the internet,contributes to the development and implementation of policies aimed at stopping or ameliorating the spread of diseases.In this manuscript,the role of official social network perturbations generated by public health officials to slow down or stop a disease outbreak are studied over distinct classes of static social networks.The dynamics are stochastic in nature with individuals(nodes)being assigned fixed levels of education or wealth.Nodes may change their epidemiological status from susceptible,to infected and to recovered.Most importantly,it is assumed that when the prevalence reaches a pre-determined threshold level,P*,information,called awareness in our framework,starts to spread,a process triggered by public health authorities.Information is assumed to spread over the same static network and whether or not one becomes a temporary informer,is a function of his/her level of education or wealth and epidemiological status.Stochastic simulations show that threshold selection Pand the value of the average basic reproduction number impact the final epidemic size differentially.For the ErdÖos-Rényi and Small-world networks,an optimal choice for Pthat minimize the final epidemic size can be identified under some conditions while for Scalefree networks this is not case.
