We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors.We also prove tha...We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors.We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.展开更多
The P_(k)-path graph P_(k)(G)corresponding to a graph G has for vertices the set of all paths of length k in G.Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path...The P_(k)-path graph P_(k)(G)corresponding to a graph G has for vertices the set of all paths of length k in G.Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k-1 in G,and their union forms either a cycle or a path of length k+1.Let Ek={(v,p),p E V(P_(k)(G)),v is an end vertex of p in G},we define total P_(k)-graphs T_(k)(G)as Yk(G)=(V(G)UV(P_(k)(G)),E(G)U E(PI(G))U Ek).In this note,we introduce total P,-graphs Th(G)and study their edge connectivity,as the generaliza-tion of total graphs.展开更多
Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we determine the cycle multiplicity of and then obtain the formula of cycle multiplicity of total graph of complete bi...Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we determine the cycle multiplicity of and then obtain the formula of cycle multiplicity of total graph of complete bipartite graph, this generalizes the result for, which is given by M.M. Akbar Ali in [1].展开更多
The atom-bond sum-connectivity(ABS)index,put forward by[J.Math.Chem.,2022,60(10):20812093],exhibits a strong link with the acentric factor of octane isomers.The experimental physico-chemical properties of octane isome...The atom-bond sum-connectivity(ABS)index,put forward by[J.Math.Chem.,2022,60(10):20812093],exhibits a strong link with the acentric factor of octane isomers.The experimental physico-chemical properties of octane isomers,such as boiling point,of formation are found to be better measured by the ABS index than by the Randi,atom-bond connectivity(ABC),and sum-connectivity(SC)indices.One important source of information for researching the molecular structure is the bounds for its topological indices.The extrema of the ABS index of the line,total,and Mycielski graphs are calculated in this work.Moreover,the pertinent extremal graphs were illustrated.展开更多
LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl（G）=Δand the list total chromatic number...LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl（G）=Δand the list total chromatic numberχl（G）=Δ＋1.展开更多
Non-intrusive load monitoring is a technique for monitoring the operating conditions of electrical appliances by collecting the aggregated electrical information at the household power inlet.Despite several studies on...Non-intrusive load monitoring is a technique for monitoring the operating conditions of electrical appliances by collecting the aggregated electrical information at the household power inlet.Despite several studies on the mining of unique load characteristics,few studies have extensively considered the high computational burden and sample training.Based on lowfrequency sampling data,a non-intrusive load monitoring algorithm utilizing the graph total variation(GTV)is proposed in this study.The algorithm can effectively depict the load state without the need for prior training.First,the combined Kmeans clustering algorithm and graph signals are used to build concise and accurate graph structures as load models.The GTV representing the internal structure of the graph signal is introduced as the optimization model and solved using the augmented Lagrangian iterative algorithm.The introduction of the difference operator reduces the computing cost and addresses the inaccurate reconstruction of the graph signal.With low-frequency sampling data,the algorithm only requires a little prior data and no training,thereby reducing the computing cost.Experiments conducted using the reference energy disaggregation dataset and almanac of minutely power dataset demonstrated the stable superiority of the algorithm and its low computational burden.展开更多
A simple graph G=(V,E)is said to be vertex Euclidean if there exists a bijection f from V to{1,2,...,|V|}such that f(u)+f(v)>f(w)for each C3 subgraph with vertex set{u,v,w},where f(u)<f(v)<f(w).The vertex Euc...A simple graph G=(V,E)is said to be vertex Euclidean if there exists a bijection f from V to{1,2,...,|V|}such that f(u)+f(v)>f(w)for each C3 subgraph with vertex set{u,v,w},where f(u)<f(v)<f(w).The vertex Euclidean deficiency of a graph G,denotedμv Euclid(G),is the smallest positive integer n such that G∪N_(n) is vertex Euclidean.In this paper,we introduce some methods for deriving the vertex Euclidean properties of some simple graphs.展开更多
基金the financial support from the Slovenian Research Agency(research core funding No.P1-0222).
文摘We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors.We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.
基金supported by Natural Sciences Foundation of Guangxi Province(2012GXNSFBA053005)
文摘The P_(k)-path graph P_(k)(G)corresponding to a graph G has for vertices the set of all paths of length k in G.Two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k-1 in G,and their union forms either a cycle or a path of length k+1.Let Ek={(v,p),p E V(P_(k)(G)),v is an end vertex of p in G},we define total P_(k)-graphs T_(k)(G)as Yk(G)=(V(G)UV(P_(k)(G)),E(G)U E(PI(G))U Ek).In this note,we introduce total P,-graphs Th(G)and study their edge connectivity,as the generaliza-tion of total graphs.
文摘Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we determine the cycle multiplicity of and then obtain the formula of cycle multiplicity of total graph of complete bipartite graph, this generalizes the result for, which is given by M.M. Akbar Ali in [1].
基金Supported by Ningbo NSF(No.2021J234)Zhejiang Provincial Philosophy and Social Sciences Planning Project(No.24NDJC057YB)。
文摘The atom-bond sum-connectivity(ABS)index,put forward by[J.Math.Chem.,2022,60(10):20812093],exhibits a strong link with the acentric factor of octane isomers.The experimental physico-chemical properties of octane isomers,such as boiling point,of formation are found to be better measured by the ABS index than by the Randi,atom-bond connectivity(ABC),and sum-connectivity(SC)indices.One important source of information for researching the molecular structure is the bounds for its topological indices.The extrema of the ABS index of the line,total,and Mycielski graphs are calculated in this work.Moreover,the pertinent extremal graphs were illustrated.
基金Supported by National Natural Science Foundation of China(Grant Nos.11201440 and 11271006)Graduate Independent Innovation Foundation of Shandong University(Grant No.yzc12100)
文摘LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl（G）=Δand the list total chromatic numberχl（G）=Δ＋1.
基金supported by National Natural Science Foundation of China(No.52107117)。
文摘Non-intrusive load monitoring is a technique for monitoring the operating conditions of electrical appliances by collecting the aggregated electrical information at the household power inlet.Despite several studies on the mining of unique load characteristics,few studies have extensively considered the high computational burden and sample training.Based on lowfrequency sampling data,a non-intrusive load monitoring algorithm utilizing the graph total variation(GTV)is proposed in this study.The algorithm can effectively depict the load state without the need for prior training.First,the combined Kmeans clustering algorithm and graph signals are used to build concise and accurate graph structures as load models.The GTV representing the internal structure of the graph signal is introduced as the optimization model and solved using the augmented Lagrangian iterative algorithm.The introduction of the difference operator reduces the computing cost and addresses the inaccurate reconstruction of the graph signal.With low-frequency sampling data,the algorithm only requires a little prior data and no training,thereby reducing the computing cost.Experiments conducted using the reference energy disaggregation dataset and almanac of minutely power dataset demonstrated the stable superiority of the algorithm and its low computational burden.
文摘A simple graph G=(V,E)is said to be vertex Euclidean if there exists a bijection f from V to{1,2,...,|V|}such that f(u)+f(v)>f(w)for each C3 subgraph with vertex set{u,v,w},where f(u)<f(v)<f(w).The vertex Euclidean deficiency of a graph G,denotedμv Euclid(G),is the smallest positive integer n such that G∪N_(n) is vertex Euclidean.In this paper,we introduce some methods for deriving the vertex Euclidean properties of some simple graphs.
