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《Discrete Mathematics》2022,345(1):112669
In this paper, we consider two kinds of spectral extremal questions. The first asks which graph attains the maximum Q-index over all graphs of order n and size ? The second asks which graph attains the maximum Q-index over all -bipartite graphs with edges? We solve the first question for , and the second question for . The maximum Q-index on connected -bipartite graphs is also determined for . 相似文献
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《Discrete Mathematics》2022,345(4):112774
Chvátal and Erdös (1972) [5] proved that, for a k-connected graph G, if the stability number , then G is Hamilton-connected () or Hamiltonian () or traceable (). Motivated by the result, we focus on tight sufficient spectral conditions for k-connected graphs to possess Hamiltonian s-properties. We say that a graph possesses Hamiltonian s-properties, which means that the graph is Hamilton-connected if , Hamiltonian if , and traceable if .For a real number , and for a k-connected graph G with order n, degree diagonal matrix and adjacency matrix , we have identified best possible upper bounds for the spectral radius , where Γ is either G or the complement of G, to warrant that G possesses Hamiltonian s-properties. Sufficient conditions for a graph G to possess Hamiltonian s-properties in terms of upper bounds for the Laplacian spectral radius as well as lower bounds of the algebraic connectivity of G are also obtained. Other best possible spectral conditions for Hamiltonian s-properties are also discussed. 相似文献
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《Discrete Mathematics》2022,345(5):112802
We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time , the vertex is introduced together with m edges joining the new vertex with m different vertices chosen uniformly at random from . We prove that this random graph obeys convergence law for first-order sentences with at most variables. 相似文献
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《Discrete Mathematics》2020,343(12):112117
Let be an edge-colored graph of order . The minimum color degree of , denoted by , is the largest integer such that for every vertex , there are at least distinct colors on edges incident to . We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if , then contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if and , then contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if , then contains vertex-disjoint rainbow triangles. For any integer , we show that if and , then contains vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of edge-disjoint rainbow triangles. 相似文献
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《Discrete Mathematics》2022,345(5):112786
Let G be a connected graph with vertices and edges. The nullity of G, denoted by , is the multiplicity of eigenvalue zero of the adjacency matrix of G. Ma, Wong and Tian (2016) proved that unless G is a cycle of order a multiple of 4, where is the elementary cyclic number of G and is the number of leaves of G. Recently, Chang, Chang and Zheng (2020) characterized the leaf-free graphs with nullity , thus leaving the problem to characterize connected graphs G with nullity when . In this paper, we solve this problem completely. 相似文献
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《Discrete Mathematics》2023,346(4):113304
In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size ? In this note, we improve upon a recent result of Frankl and resolve this problem for and . 相似文献
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The k-subset sum problem over finite fields is a classical NP-complete problem. Motivated by coding theory applications, a more complex problem is the higher m-th moment k-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the m-th moment k-subset sum problem over finite fields for each fixed m when the evaluation set is the image set of a monomial or Dickson polynomial of any degree n. In the classical case , this recovers previous results of Nguyen-Wang (the case ) [22] and the results of Choe-Choe (the case ) [3]. 相似文献
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《Discrete Mathematics》2020,343(2):111679
A path in an edge-colored graph is called monochromatic if any two edges on the path have the same color. For , an edge-colored graph is said to be monochromatic -edge-connected if every two distinct vertices of are connected by at least edge-disjoint monochromatic paths, and is said to be uniformly monochromatic -edge-connected if every two distinct vertices are connected by at least edge-disjoint monochromatic paths such that all edges of these paths are colored with a same color. We use and to denote the maximum number of colors that ensures to be monochromatic -edge-connected and, respectively, to be uniformly monochromatic -edge-connected. In this paper, we first conjecture that for any -edge-connected graph , , where is a minimum -edge-connected spanning subgraph of . We verify the conjecture for . We also prove the conjecture for and with . When is a minimal -edge-connected graph, we give an upper bound of , i.e., . For the uniformly monochromatic -edge-connectivity, we prove that for all , , where is a minimum -edge-connected spanning subgraph of . 相似文献
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《Discrete Mathematics》2022,345(3):112731
Let be the matching number of a graph G. A characterization of the graphs with given maximum odd degree and smallest possible matching number is given by Henning and Shozi (2021) [13]. In this paper we complete our study by giving a characterization of the graphs with given maximum even degree and smallest possible matching number. In 2018 Henning and Yeo [10] proved that if G is a connected graph of order n, size m and maximum degree k where is even, then , unless G is k-regular and . In this paper, we give a complete characterization of the graphs that achieve equality in this bound when the maximum degree k is even, thereby completing our study of graphs with given maximum degree and smallest possible matching number. 相似文献
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《Discrete Mathematics》2021,344(12):112604
A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index of G is Δ or . A graph G is class 1 if , and class 2 if ; G is Δ-critical if it is connected, class 2 and for every . A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, for every . Such graphs have intimate relation to -co-critical graphs, where a non-complete graph G is -co-critical if there exists a k-coloring of such that G does not contain a monochromatic copy of but every k-coloring of contains a monochromatic copy of for every . We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all -co-critical graphs. We prove that if G is a -co-critical graph on vertices, then where ε is the remainder of when divided by 2. This bound is best possible for all and . 相似文献
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《Discrete Mathematics》2022,345(12):113083
Let G be a graph, the order of G, the connectivity of G and k a positive integer such that . Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A Hamiltonian path of a graph G is a spanning path of G. A bipartite graph G with vertex sets and is defined to be Hamiltonian-laceable if such that and for every pair of vertices and , there exists a Hamiltonian path in G with endpoints p and q, or and for every pair of vertices , there exists a Hamiltonian path in G with endpoints p and q. Let G be a bipartite graph with bipartition . Define to be a maximum integer such that and (1) for each non-empty subset S of X, if , then , and if , then ; and (2) for each non-empty subset S of Y, if , then , and if , then ; and (3) if there is no non-negative integer satisfying (1) and (2).Let G be a bipartite graph with bipartition such that and . In this paper, we show that if , then G is Hamiltonian-laceable; or if , then for every pair of vertices and , there is an -path P in G with . We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs. 相似文献
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《Discrete Mathematics》2022,345(12):113078
Let G be a simple connected graph and let be the sum of the first k largest Laplacian eigenvalues of G. It was conjectured by Brouwer in 2006 that holds for . The case was proved by Haemers, Mohammadian and Tayfeh-Rezaie [Linear Algebra Appl., 2010]. In this paper, we propose the full Brouwer's Laplacian spectrum conjecture and we prove the conjecture holds for which also confirm the conjecture of Guan et al. in 2014. 相似文献
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