In 2012, Ponraj et al. defined a concept of k-product cordial labeling as follows: Let f be a map from V(G)to { 0,1,⋯,k−1 }where k is an integer, 1≤k≤| V(G) |. For each edge uvassign the label f(u)f(v)(modk). f is c...In 2012, Ponraj et al. defined a concept of k-product cordial labeling as follows: Let f be a map from V(G)to { 0,1,⋯,k−1 }where k is an integer, 1≤k≤| V(G) |. For each edge uvassign the label f(u)f(v)(modk). f is called a k-product cordial labeling if | vf(i)−vf(j) |≤1, and | ef(i)−ef(j) |≤1, i,j∈{ 0,1,⋯,k−1 }, where vf(x)and ef(x)denote the number of vertices and edges respectively labeled with x (x=0,1,⋯,k−1). Motivated by this concept, we further studied and established that several families of graphs admit k-product cordial labeling. In this paper, we show that the path graphs Pnadmit k-product cordial labeling.展开更多
This paper, an addendum to “Dialectical Thermodynamics’ solution to the conceptual imbroglio that is the reversible path”, this journal, 10, 775-799, was written in response to the requests of several readers to pr...This paper, an addendum to “Dialectical Thermodynamics’ solution to the conceptual imbroglio that is the reversible path”, this journal, 10, 775-799, was written in response to the requests of several readers to provide further evidence of the said “imbroglio”. The evidence here presented relates to the incompatibility existing between the total-entropy and the Gibbs energy prescriptions for the reversible path. The previously published proof of the negentropic nature of the transformation of heat into work is here included to validate out conclusions about the Gibbs energy perspective.展开更多
文摘In 2012, Ponraj et al. defined a concept of k-product cordial labeling as follows: Let f be a map from V(G)to { 0,1,⋯,k−1 }where k is an integer, 1≤k≤| V(G) |. For each edge uvassign the label f(u)f(v)(modk). f is called a k-product cordial labeling if | vf(i)−vf(j) |≤1, and | ef(i)−ef(j) |≤1, i,j∈{ 0,1,⋯,k−1 }, where vf(x)and ef(x)denote the number of vertices and edges respectively labeled with x (x=0,1,⋯,k−1). Motivated by this concept, we further studied and established that several families of graphs admit k-product cordial labeling. In this paper, we show that the path graphs Pnadmit k-product cordial labeling.
文摘This paper, an addendum to “Dialectical Thermodynamics’ solution to the conceptual imbroglio that is the reversible path”, this journal, 10, 775-799, was written in response to the requests of several readers to provide further evidence of the said “imbroglio”. The evidence here presented relates to the incompatibility existing between the total-entropy and the Gibbs energy prescriptions for the reversible path. The previously published proof of the negentropic nature of the transformation of heat into work is here included to validate out conclusions about the Gibbs energy perspective.
