A graph G is said to be determined by its Laplacian spectrum if any graph having the same Laplacian spectrum as G is isomorphic to G.We consider θ-graphs,that is,graphs obtained by subdividing the edges of the multigraph consist of three parallel edges.In this paper,some special θ-graphs are determined by their Laplacian spectra.
Nuclear reaction rate A is a significant factor in processes of nucleosyntheses.A multi-layer directed-weighted nuclear reaction network,in which the reaction rate is taken as the weight,and neutron,proton,4 He and the remainder nuclei as the criteria for different reaction layers,is for the first time built based on all thermonuclear reactions in the JINA REACLIB database.Our results show that with the increase in the stellar temperature T9,the distribution of nuclear reaction rates on the R-layer network demonstrates a transition from unimodal to bimodal distributions.Nuclei on the R-layer in the region of A=[1,2.5×101]have a more complicated out-going degree distribution than that in the region of A=[1011,1013],and the number of involved nuclei at T9=1 is very different from the one at T9=3.The redundant nuclei in the region of A=[1,2.5×101]at T9=3 prefer(γ,p)and(γ,α)reactions to the ones at T9=1,which produce nuclei around theβstable line.This work offers a novel way to the big-data analysis on the nuclear reaction network at stellar temperatures.
Nucleosynthesis is a complex process in astro-nuclear evolution.In this work,we constructed a directed,multi-layer,nuclear reaction network,using the substrate-product method from a thermonuclear reaction database,JINA REACLIB.The network contained four layers,namely n-,p-,h-and r-,which correspond to the types of reactions involved in neutrons,protons,~4He and the remaining atomic particles,respectively.The degree values(i.e.numbers of reactions),for three layers of n-,p-,and h-,have a significant correlation with one another,and their topological structures exhibit a similar regularity.However,the r-layer has a more complex topological structure compared to the other layers;thus,its correlation of degree values with the other three layers is not as strong as that seen in the other layers.To analyze the motif structure of the nuclear reaction network,we used the software package"mfinder".We identified the most frequent reaction patterns of the interconnections that occurred amongst the different nuclides.This work provides a novel approach to investigating the nuclear reaction network that prevails in the astrophysical context.
