A set S of vertices of a graph G is called a decycling set if G-S is acyclic.The smallest size of a decycling set is called the decycling number of G and is denoted by ∇(G).In this paper,we investigate the decycling n...A set S of vertices of a graph G is called a decycling set if G-S is acyclic.The smallest size of a decycling set is called the decycling number of G and is denoted by ∇(G).In this paper,we investigate the decycling number of type-k Halin graphs,focusing on those that are formed from trees that have just two degrees k and 3.For any type-k Halin graph G of order n,we prove that(k-2)n+k^(2)-4k+5/(k-1)^(2)≤∇(G)≤n+k-3/k-1.The result not only supports the largest forest conjecture due to Albertson and Berman(1976),but also offers a tight lower bound for the decycling number of type-3 Halin graphs and several type-k Halin graphs.Moreover,a new formula to determine the cardinality of any decycling set S of a type-k Halin graph G is provided.展开更多
A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vert...A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vertex.This concept was introduced by Tutte as an extension of face colorings,and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow,known as the 5-Flow Conjecture.This conjecture is verified for some graph classes and remains unresolved as of today.In this paper,we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow,which improves several known results.展开更多
A graph whose edges are labeled either as positive or negative is called a signed graph.Hameed et al.introduced signed distance and distance compatibility in 2021,initially to characterize balanced signed graphs which...A graph whose edges are labeled either as positive or negative is called a signed graph.Hameed et al.introduced signed distance and distance compatibility in 2021,initially to characterize balanced signed graphs which have nice spectral properties.This article mainly studies the conjecture proposed by Shijin et al.on the distance compatibility of the direct product of signed graphs,and provides necessary and sufficient conditions for the distance compatibility of the direct product of signed graphs.Some further questions regarding distance compatibility are also posed.展开更多
In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piece...In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.展开更多
A graph G is H-free,if it contains no H as a subgraph.A graph G is said to be H-minor free,if it does not contain H as a minor.In 2010,Nikiforov asked that what the maximum spectral radius of an H-free graph of order ...A graph G is H-free,if it contains no H as a subgraph.A graph G is said to be H-minor free,if it does not contain H as a minor.In 2010,Nikiforov asked that what the maximum spectral radius of an H-free graph of order n is.In this paper,we consider some Brualdi-Solheid-Turan type problems on bipartite graphs.In 2015,Zhai,Lin and Gong in[Linear Algebra Appl.,2015,471:21-27]proved that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains a C_(2k+2) unless G≌K_(k,n-k).First,we give a new and more simple proof for the above theorem.Second,we prove that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains all T_(2k+3) unless G≌K_(k,n-k).Finally,we prove that among all outerplanar bipartite graphs on n≥308026 vertices,K_(1,n-1) attains the maximum spectral radius.展开更多
The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G ...The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G is defined as ME(G) = Pni=1 |λi|. By the famous Coulson’s formula, matching energies can also be calculated by an improper integral depending on a parameter. A k-claw attaching graph Gu(k) refers to the graph obtained by attaching k pendent edges to the graph G at the vertex u, where u is called the root of Gu(k). In this paper, we use some theories of mathematical analysis to obtain a new technique to compare the matching energies of two k-claw attaching graphs Gu(k) and Hv(k) with the same order, that is, limk→∞[ME(Gu(k)) − ME(Hv(k))] = ME(G − u) − ME(H − v). By the technique, we finally determine unicyclic graphs of order n with the 9th to 13th minimal matching energies for all n ≥ 58.展开更多
A graph G is called d-degenerate if every subgraph of G has a vertex of degree at most d.It was known that planar graphs are 5-degenerate and every planar graph without k-cycles for some prescribed k∈{3,5,6}is 3-dege...A graph G is called d-degenerate if every subgraph of G has a vertex of degree at most d.It was known that planar graphs are 5-degenerate and every planar graph without k-cycles for some prescribed k∈{3,5,6}is 3-degenerate.In this paper,we show that if G is a planar graph without kites and 9-or 10-cycles,then G is 3-degenerate,hence 4-choosable and list vertex 2-arborable.展开更多
Determining the crossing number of a given graph is NP-complete. The cycle of length m is denoted by Cm = v1v2…vmv1. G^((1))_(m) (m ≥ 5) is the graph obtained from Cm by adding two edges v1v3 and vlvl+2 (3 ≤ l ≤ m...Determining the crossing number of a given graph is NP-complete. The cycle of length m is denoted by Cm = v1v2…vmv1. G^((1))_(m) (m ≥ 5) is the graph obtained from Cm by adding two edges v1v3 and vlvl+2 (3 ≤ l ≤ m−2), G^((2))m (m ≥ 4) is the graph obtained from Cm by adding two edges v1v3 and v2v4. The famous Zarankiewicz’s conjecture on the crossing number of the complete bipartite graph Km,n states that cr(Km,n)=Z(m,n)=[m/2][m-1/2][n/2[n-1/2].Based on Zarankiewicz’s conjecture, a natural problem is to study the change in the crossingnumber of the graphs obtained from the complete bipartite graph by adding certain edge sets.If Zarankiewicz’s conjecture is true, this paper proves that cr(G^((1))_(m)+Kn)=Z(m,n)+2[n/2] and cr(G^((2))_(m)+Kn)=Z(m,n)+n.展开更多
DP-coloring as a generalization of list coloring was introduced recently by Dvo˘r´ak and Postle.In this paper,we show that planar graphs without 5-cycles adjacent to two triangles are DP-4-colorable,which improve...DP-coloring as a generalization of list coloring was introduced recently by Dvo˘r´ak and Postle.In this paper,we show that planar graphs without 5-cycles adjacent to two triangles are DP-4-colorable,which improves the results of[Discrete Math.,2018,341(7):1983–1986]and[Discrete Appl.Math.,2020,277:245–251].展开更多
For a graph G,a vertex is said to be pendant if its neighborhood contains exactly one vertex.In this paper,we determine the extremal graphs among all n-vertex graphs with the minimum spectral radius andβpendant verti...For a graph G,a vertex is said to be pendant if its neighborhood contains exactly one vertex.In this paper,we determine the extremal graphs among all n-vertex graphs with the minimum spectral radius andβpendant vertices,whereβe{1,2,3,4,n-3,n-2,n-1}.展开更多
Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_...Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_(N-1)by adding anew vertexνconnecting k vertices of K_(N-1).A graph G withχ(G)=k+1 is called edge-critical if G contains an edge e such thatχ(G-e)=k.A considerable amount of research has been conducted by previous scholars on Ramsey numbers ofgraphs.In this study,we show that for an edge-critical graph G with x(G)=k+1,when k≥2,1≥2,and n is sufficiently large,R(G,K_(1)+nK_(t))=knt+1 and r,(G,K_(1)+nK_(t))=(k-1)nt+1.展开更多
For a simple graph G,let A(G)and D(G)be the adjacency matrix and the diagonal degree matrix of G,respectively.[Appl.Anal.Discrete Math.,2017,11(1):81-107]defined the matrix A_(α)(G)of G as A_(α)(G)=αD(G)(1-α)A(G)...For a simple graph G,let A(G)and D(G)be the adjacency matrix and the diagonal degree matrix of G,respectively.[Appl.Anal.Discrete Math.,2017,11(1):81-107]defined the matrix A_(α)(G)of G as A_(α)(G)=αD(G)(1-α)A(G),α∈[0,1].The Aa-spectral radius is the largest eigenvalue of A_(α)(G).Let G_(n,β) be the set graphs with order n and dissociation numberβ.In this paper,we identify the b with maximal A_(α)-spectral radius among all graphs in G_(n,β).展开更多
A dominating induced matching(DIM)of G is an induced matching that dominates every edge of G.In this note,we completely determine the number of DIMs in the generalized Petersen graph P(n,k).We prove that if P(n,k)is a...A dominating induced matching(DIM)of G is an induced matching that dominates every edge of G.In this note,we completely determine the number of DIMs in the generalized Petersen graph P(n,k).We prove that if P(n,k)is a generalized Petersen graph with n=0(mod 5)and k=2,3(mod 5),then E(P(n,k))can be partitioned into five DIMs.Meanwhile,in the left cases k=0,1,4(mod 5),we build some counterexamples to show that there exist some P(n,k)'s which are DIM-free.展开更多
The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ...The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.展开更多
A kernel in a directed graph D=(V,A)is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V\K there is a vertex u in K,such that(v,u)is an arc of D.It is well known that the...A kernel in a directed graph D=(V,A)is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V\K there is a vertex u in K,such that(v,u)is an arc of D.It is well known that the problem of the existence of a kernel is NP-complete for a general digraph.Bang-Jensen and Gutin pose an interesting problem(Problem 12.3.5)in their book[Digraphs:Theory,Algorithms and Applications,London:Springer-Verlag,2000]:to characterize all circular digraphs with kernels.In this paper,we study the problem of the existence of the kernel for several special classes of circular digraphs.Moreover,a class of counterexamples is given for the Duchet kernel conjecture(for every connected kernel-less digraph which is not an odd directed cycle,there exists an arc which can be removed and the obtained digraph is still kernel-less).展开更多
Graphs have been widely used in fields ranging from chemical informatics to social network analysis.Graph-related problems become increasingly significant,with subgraph matching standing out as one of the most challen...Graphs have been widely used in fields ranging from chemical informatics to social network analysis.Graph-related problems become increasingly significant,with subgraph matching standing out as one of the most challenging tasks.The goal of subgraph matching is to find all subgraphs in the data graph that are isomorphic to the query graph.Traditional methods mostly rely on search strategies with high computational complexity and are hard to apply to large-scale real datasets.With the advent of graph neural networks(GNNs),researchers have turned to GNNs to address subgraph matching problems.However,the multi-attributed features on nodes and edges are overlooked during the learning of graphs,which causes inaccurate results in real-world scenarios.To tackle this problem,we propose a novel model called subgraph matching on multi-attributed graph network(SGMAN).SGMAN first utilizes improved line graphs to capture node and edge features.Then,SGMAN integrates GNN and contrastive learning(CL)to derive graph representation embeddings and calculate the matching matrix to represent the matching results.We conduct experiments on public datasets,and the results affirm the superior performance of our model.展开更多
The Pfaffian property of graphs is of fundamental importance in graph theory,as it precisely characterizes those graphs for which the number of perfect matchings can be computed in polynomial time with respect to the ...The Pfaffian property of graphs is of fundamental importance in graph theory,as it precisely characterizes those graphs for which the number of perfect matchings can be computed in polynomial time with respect to the number of edges.The study of Pfaffian graphs originated from the enumeration of perfect matching in planar graphs.References[5,6,8]demonstrated that every planar graph is Pfaffian.Therefore,the Pfaffian property and planarity of graphs play a vital role in modern matching theory.This paper contributes a complete characterization of the Pfaffian property and planarity of connected Cayley graphs over the dicyclic group T_(4n) of order 4n(n≥3),shows that the Cayley graph Cay(T_(4n),S)is Pfaffian if and only if n is odd and S={a^(k_(1)),a^(2n−k_(1)),ba^(k_(2)),ba^(n+k_(2))},where 1≤k_(1)≤n−1,0≤k_(2)≤n−1 and(k_(1),n)=1,and furthermore,shows that Cay(T4n,S)is never planar.展开更多
Let R be afinite commutative ring with identity 1.The U-clean graph of R,denoted by U-Cl(R),is a graph with vertices in form(e,u),where e is a nonzero idempotent of R and u is a unit of R.In this paper,some basic prope...Let R be afinite commutative ring with identity 1.The U-clean graph of R,denoted by U-Cl(R),is a graph with vertices in form(e,u),where e is a nonzero idempotent of R and u is a unit of R.In this paper,some basic properties of U-Cl(R)and the explicit structures of U-Cl(Zp×Zq)are given,where p,q are primes.We prove that U-Cl(Zp×Zq)is Eulerlian if and only if p=2,q=2.Moreover,the clique number,the chromatic number of the U-clean graph for some classes of rings are given in this paper.展开更多
A Gallai k-coloring is a k-edge-coloring of a complete graph in which there are no rainbow triangles.For given graphs G_(1),G_(2),G_(3)and nonnegative integers r,s,t with k=r+s+t,the k-colored Gallai-Ramsey number grk...A Gallai k-coloring is a k-edge-coloring of a complete graph in which there are no rainbow triangles.For given graphs G_(1),G_(2),G_(3)and nonnegative integers r,s,t with k=r+s+t,the k-colored Gallai-Ramsey number grk(K_(3):r·G_(1),s·G_(2),t·G_(3))is the minimum integer n such that every Gallai k-colored Kncontains a monochromatic copy of G_(1)colored by one of the first r colors or a monochromatic copy of G_(2)colored by one of the middle s colors or a monochromatic copy of G_(3)colored by one of the last t colors.In this paper,we determine the value of GallaiRamsey number in the case that G_(1)=B_(3)^(+),G_(2)=S_(3)^(+)and G_(3)=K_(3).Then the Gallai-Ramsey numbers grk(K_(3):B_(3)^(+)),grk(K_(3):S_(3)^(+))and grk(K_(3):K_(3))are obtained,respectively.Furthermore,the Gallai-Ramsey numbers grk(K_(3):r·B_(3)^(+),(k-r)·S_(3)^(+)),grk(K_(3):r·B_(3)^(+),(k-r)·K_(3))and grk(K_(3):s·S_(3)^(+),(k-s)·K_(3))are obtained,respectively.展开更多
Knowledge graphs (KGs) offer a structured, machine-readable format for organizing complex information. In heterogeneous catalysis, where data on catalytic materials, reaction conditions, mechanisms, and synthesis rout...Knowledge graphs (KGs) offer a structured, machine-readable format for organizing complex information. In heterogeneous catalysis, where data on catalytic materials, reaction conditions, mechanisms, and synthesis routes are dispersed across diverse sources, KGs provide a semantic framework that supports data integration under the FAIR (Findable, Accessible, Interoperable, and Reusable) principles. This review aims to survey recent developments in catalysis KGs, describe the main techniques for graph construction, and highlight how artificial intelligence, particularly large language models (LLMs), enhances graph generation and query. We conducted a systematic analysis of the literature, focusing on ontology-guided text mining pipelines, graph population methods, and maintenance strategies. Our review identifies key trends: ontology-based approaches enable the automated extraction of domain knowledge, LLM-driven retrieval-augmented generation supports natural-language queries, and scalable graph architectures range from a few thousand to over a million triples. We discuss state-of-the-art applications, such as catalyst recommendation systems and reaction mechanism discovery tools, and examine the major challenges, including data heterogeneity, ontology alignment, and long-term graph curation. We conclude that KGs, when combined with AI methods, hold significant promise for accelerating catalyst discovery and knowledge management, but progress depends on establishing community standards for ontology development and maintenance. This review provides a roadmap for researchers seeking to leverage KGs to advance heterogeneous catalysis research.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.11171114,11401576)Hotan Prefecture Science and Technology Bureau General Project(Grant No.20220212)。
文摘A set S of vertices of a graph G is called a decycling set if G-S is acyclic.The smallest size of a decycling set is called the decycling number of G and is denoted by ∇(G).In this paper,we investigate the decycling number of type-k Halin graphs,focusing on those that are formed from trees that have just two degrees k and 3.For any type-k Halin graph G of order n,we prove that(k-2)n+k^(2)-4k+5/(k-1)^(2)≤∇(G)≤n+k-3/k-1.The result not only supports the largest forest conjecture due to Albertson and Berman(1976),but also offers a tight lower bound for the decycling number of type-3 Halin graphs and several type-k Halin graphs.Moreover,a new formula to determine the cardinality of any decycling set S of a type-k Halin graph G is provided.
文摘A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vertex.This concept was introduced by Tutte as an extension of face colorings,and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow,known as the 5-Flow Conjecture.This conjecture is verified for some graph classes and remains unresolved as of today.In this paper,we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow,which improves several known results.
基金Supported by the National Natural Science Foundation of China(Grant No.12071260)。
文摘A graph whose edges are labeled either as positive or negative is called a signed graph.Hameed et al.introduced signed distance and distance compatibility in 2021,initially to characterize balanced signed graphs which have nice spectral properties.This article mainly studies the conjecture proposed by Shijin et al.on the distance compatibility of the direct product of signed graphs,and provides necessary and sufficient conditions for the distance compatibility of the direct product of signed graphs.Some further questions regarding distance compatibility are also posed.
基金supported in part by the NSFC(11801496,11926352)the Fok Ying-Tung Education Foundation(China)the Hubei Key Laboratory of Applied Mathematics(Hubei University).
文摘In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.
基金Supported by NSFC(No.12271162)Natural Science Foundation of Shanghai(No.22ZR1416300).
文摘A graph G is H-free,if it contains no H as a subgraph.A graph G is said to be H-minor free,if it does not contain H as a minor.In 2010,Nikiforov asked that what the maximum spectral radius of an H-free graph of order n is.In this paper,we consider some Brualdi-Solheid-Turan type problems on bipartite graphs.In 2015,Zhai,Lin and Gong in[Linear Algebra Appl.,2015,471:21-27]proved that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains a C_(2k+2) unless G≌K_(k,n-k).First,we give a new and more simple proof for the above theorem.Second,we prove that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains all T_(2k+3) unless G≌K_(k,n-k).Finally,we prove that among all outerplanar bipartite graphs on n≥308026 vertices,K_(1,n-1) attains the maximum spectral radius.
基金Supported by the National Natural Science Foundation of China(Nos.12271439,11871398)the National College Students Innovation and Entrepreneurship Training Program(No.201910699173)。
文摘The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G is defined as ME(G) = Pni=1 |λi|. By the famous Coulson’s formula, matching energies can also be calculated by an improper integral depending on a parameter. A k-claw attaching graph Gu(k) refers to the graph obtained by attaching k pendent edges to the graph G at the vertex u, where u is called the root of Gu(k). In this paper, we use some theories of mathematical analysis to obtain a new technique to compare the matching energies of two k-claw attaching graphs Gu(k) and Hv(k) with the same order, that is, limk→∞[ME(Gu(k)) − ME(Hv(k))] = ME(G − u) − ME(H − v). By the technique, we finally determine unicyclic graphs of order n with the 9th to 13th minimal matching energies for all n ≥ 58.
文摘A graph G is called d-degenerate if every subgraph of G has a vertex of degree at most d.It was known that planar graphs are 5-degenerate and every planar graph without k-cycles for some prescribed k∈{3,5,6}is 3-degenerate.In this paper,we show that if G is a planar graph without kites and 9-or 10-cycles,then G is 3-degenerate,hence 4-choosable and list vertex 2-arborable.
基金Supported by Changsha Natural Science Foundation(No.kq2208001)the Key Project Funded by Hunan Provincial Department of Education(No.21A0590)。
文摘Determining the crossing number of a given graph is NP-complete. The cycle of length m is denoted by Cm = v1v2…vmv1. G^((1))_(m) (m ≥ 5) is the graph obtained from Cm by adding two edges v1v3 and vlvl+2 (3 ≤ l ≤ m−2), G^((2))m (m ≥ 4) is the graph obtained from Cm by adding two edges v1v3 and v2v4. The famous Zarankiewicz’s conjecture on the crossing number of the complete bipartite graph Km,n states that cr(Km,n)=Z(m,n)=[m/2][m-1/2][n/2[n-1/2].Based on Zarankiewicz’s conjecture, a natural problem is to study the change in the crossingnumber of the graphs obtained from the complete bipartite graph by adding certain edge sets.If Zarankiewicz’s conjecture is true, this paper proves that cr(G^((1))_(m)+Kn)=Z(m,n)+2[n/2] and cr(G^((2))_(m)+Kn)=Z(m,n)+n.
基金Partially supported by NSFC(No.12301436)NSF of Guangxi Province(No.2025GXNSFAA069811)。
文摘DP-coloring as a generalization of list coloring was introduced recently by Dvo˘r´ak and Postle.In this paper,we show that planar graphs without 5-cycles adjacent to two triangles are DP-4-colorable,which improves the results of[Discrete Math.,2018,341(7):1983–1986]and[Discrete Appl.Math.,2020,277:245–251].
文摘For a graph G,a vertex is said to be pendant if its neighborhood contains exactly one vertex.In this paper,we determine the extremal graphs among all n-vertex graphs with the minimum spectral radius andβpendant vertices,whereβe{1,2,3,4,n-3,n-2,n-1}.
基金supported by the National Key Research and Development Program of China(2023YFA1010200,2020YFA0713100)the National Natural Science Foundation of China(12071453)the Innovation Program for Quantum Science and Technology(2021ZD0302902).
文摘Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_(N-1)by adding anew vertexνconnecting k vertices of K_(N-1).A graph G withχ(G)=k+1 is called edge-critical if G contains an edge e such thatχ(G-e)=k.A considerable amount of research has been conducted by previous scholars on Ramsey numbers ofgraphs.In this study,we show that for an edge-critical graph G with x(G)=k+1,when k≥2,1≥2,and n is sufficiently large,R(G,K_(1)+nK_(t))=knt+1 and r,(G,K_(1)+nK_(t))=(k-1)nt+1.
基金Supported by NSFC (Nos.12171089,12271235)NSF of Jiangsu (No.BK20190919)NSF of Fujian (No.2021J02048)。
文摘For a simple graph G,let A(G)and D(G)be the adjacency matrix and the diagonal degree matrix of G,respectively.[Appl.Anal.Discrete Math.,2017,11(1):81-107]defined the matrix A_(α)(G)of G as A_(α)(G)=αD(G)(1-α)A(G),α∈[0,1].The Aa-spectral radius is the largest eigenvalue of A_(α)(G).Let G_(n,β) be the set graphs with order n and dissociation numberβ.In this paper,we identify the b with maximal A_(α)-spectral radius among all graphs in G_(n,β).
基金Ming Chen was supported by National Key Research and Development Program of China(No.2024YFA1013900)。
文摘A dominating induced matching(DIM)of G is an induced matching that dominates every edge of G.In this note,we completely determine the number of DIMs in the generalized Petersen graph P(n,k).We prove that if P(n,k)is a generalized Petersen graph with n=0(mod 5)and k=2,3(mod 5),then E(P(n,k))can be partitioned into five DIMs.Meanwhile,in the left cases k=0,1,4(mod 5),we build some counterexamples to show that there exist some P(n,k)'s which are DIM-free.
文摘The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.
文摘A kernel in a directed graph D=(V,A)is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V\K there is a vertex u in K,such that(v,u)is an arc of D.It is well known that the problem of the existence of a kernel is NP-complete for a general digraph.Bang-Jensen and Gutin pose an interesting problem(Problem 12.3.5)in their book[Digraphs:Theory,Algorithms and Applications,London:Springer-Verlag,2000]:to characterize all circular digraphs with kernels.In this paper,we study the problem of the existence of the kernel for several special classes of circular digraphs.Moreover,a class of counterexamples is given for the Duchet kernel conjecture(for every connected kernel-less digraph which is not an odd directed cycle,there exists an arc which can be removed and the obtained digraph is still kernel-less).
文摘Graphs have been widely used in fields ranging from chemical informatics to social network analysis.Graph-related problems become increasingly significant,with subgraph matching standing out as one of the most challenging tasks.The goal of subgraph matching is to find all subgraphs in the data graph that are isomorphic to the query graph.Traditional methods mostly rely on search strategies with high computational complexity and are hard to apply to large-scale real datasets.With the advent of graph neural networks(GNNs),researchers have turned to GNNs to address subgraph matching problems.However,the multi-attributed features on nodes and edges are overlooked during the learning of graphs,which causes inaccurate results in real-world scenarios.To tackle this problem,we propose a novel model called subgraph matching on multi-attributed graph network(SGMAN).SGMAN first utilizes improved line graphs to capture node and edge features.Then,SGMAN integrates GNN and contrastive learning(CL)to derive graph representation embeddings and calculate the matching matrix to represent the matching results.We conduct experiments on public datasets,and the results affirm the superior performance of our model.
基金supported by NSFC(No.12201202)NSF of Hunan Province(No.2023JJ30180)NSFC(No.12471022)。
文摘The Pfaffian property of graphs is of fundamental importance in graph theory,as it precisely characterizes those graphs for which the number of perfect matchings can be computed in polynomial time with respect to the number of edges.The study of Pfaffian graphs originated from the enumeration of perfect matching in planar graphs.References[5,6,8]demonstrated that every planar graph is Pfaffian.Therefore,the Pfaffian property and planarity of graphs play a vital role in modern matching theory.This paper contributes a complete characterization of the Pfaffian property and planarity of connected Cayley graphs over the dicyclic group T_(4n) of order 4n(n≥3),shows that the Cayley graph Cay(T_(4n),S)is Pfaffian if and only if n is odd and S={a^(k_(1)),a^(2n−k_(1)),ba^(k_(2)),ba^(n+k_(2))},where 1≤k_(1)≤n−1,0≤k_(2)≤n−1 and(k_(1),n)=1,and furthermore,shows that Cay(T4n,S)is never planar.
基金Supported by Anhui Provincial Natural Science Foundation(2008085MA06)the Key Project of Anhui Education Committee(gxyqZD2019009)。
文摘Let R be afinite commutative ring with identity 1.The U-clean graph of R,denoted by U-Cl(R),is a graph with vertices in form(e,u),where e is a nonzero idempotent of R and u is a unit of R.In this paper,some basic properties of U-Cl(R)and the explicit structures of U-Cl(Zp×Zq)are given,where p,q are primes.We prove that U-Cl(Zp×Zq)is Eulerlian if and only if p=2,q=2.Moreover,the clique number,the chromatic number of the U-clean graph for some classes of rings are given in this paper.
基金Supported by the National Natural Science Foundation of China(12161073)。
文摘A Gallai k-coloring is a k-edge-coloring of a complete graph in which there are no rainbow triangles.For given graphs G_(1),G_(2),G_(3)and nonnegative integers r,s,t with k=r+s+t,the k-colored Gallai-Ramsey number grk(K_(3):r·G_(1),s·G_(2),t·G_(3))is the minimum integer n such that every Gallai k-colored Kncontains a monochromatic copy of G_(1)colored by one of the first r colors or a monochromatic copy of G_(2)colored by one of the middle s colors or a monochromatic copy of G_(3)colored by one of the last t colors.In this paper,we determine the value of GallaiRamsey number in the case that G_(1)=B_(3)^(+),G_(2)=S_(3)^(+)and G_(3)=K_(3).Then the Gallai-Ramsey numbers grk(K_(3):B_(3)^(+)),grk(K_(3):S_(3)^(+))and grk(K_(3):K_(3))are obtained,respectively.Furthermore,the Gallai-Ramsey numbers grk(K_(3):r·B_(3)^(+),(k-r)·S_(3)^(+)),grk(K_(3):r·B_(3)^(+),(k-r)·K_(3))and grk(K_(3):s·S_(3)^(+),(k-s)·K_(3))are obtained,respectively.
基金support from the Full Bridge Fellowship for enabling the research stay at Virginia Tech.H.Xin acknowledge the financial support from the US Department of Energy,Office of Basic Energy Sciences under contract no.DE-SC0023323from the National Science Foundation through the grant 2245402 from CBET Catalysis and CDS&E programs.
文摘Knowledge graphs (KGs) offer a structured, machine-readable format for organizing complex information. In heterogeneous catalysis, where data on catalytic materials, reaction conditions, mechanisms, and synthesis routes are dispersed across diverse sources, KGs provide a semantic framework that supports data integration under the FAIR (Findable, Accessible, Interoperable, and Reusable) principles. This review aims to survey recent developments in catalysis KGs, describe the main techniques for graph construction, and highlight how artificial intelligence, particularly large language models (LLMs), enhances graph generation and query. We conducted a systematic analysis of the literature, focusing on ontology-guided text mining pipelines, graph population methods, and maintenance strategies. Our review identifies key trends: ontology-based approaches enable the automated extraction of domain knowledge, LLM-driven retrieval-augmented generation supports natural-language queries, and scalable graph architectures range from a few thousand to over a million triples. We discuss state-of-the-art applications, such as catalyst recommendation systems and reaction mechanism discovery tools, and examine the major challenges, including data heterogeneity, ontology alignment, and long-term graph curation. We conclude that KGs, when combined with AI methods, hold significant promise for accelerating catalyst discovery and knowledge management, but progress depends on establishing community standards for ontology development and maintenance. This review provides a roadmap for researchers seeking to leverage KGs to advance heterogeneous catalysis research.
