Results 11 to 20 of about 34,901 (303)
The Spectrum of Triangle-Free Graphs
SIAM Journal on Discrete Mathematics, 2023 Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) \leq \frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) \leq \frac{15n}{94}< \frac{4n}{25}$.
József Balogh +4 more
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The triangle graph $T_6$ is not SPN
The Electronic Journal of Linear Algebra, 2020 A real symmetric matrix $A$ is copositive if $x'Ax \geq 0$ for every nonnegative vector $x$. A matrix is SPN if it is a sum of a real positive semidefinite matrix and a nonnegative matrix. Every SPN matrix is copositive, but the converse does not hold for matrices of order greater than $4$.
Drury , Stephen
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On the independence number of intersection graphs of axis-parallel segments
Journal of Computational Geometry, 2023 We prove that for any triangle-free intersection graph of $n$ axis-parallel line segments in the plane, the independence number $\alpha$ of this graph is at least $\alpha \ge n/4 + \Omega(\sqrt{n})$.
Marco Caoduro +3 more
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Bounds for the augmented Zagreb index
Theory and Applications of Graphs, 2023 The augmented Zagreb index (\rm{AZI} for short) of a graph $G$, introduced by Furtula et al. in 2010, is defined as ${\rm AZI}(G)=\sum\limits_{v_iv_j\in E(G)}{\left(\frac{d(v_i)d(v_j)}{d(v_i)+d(v_j)-2}\right)}^3$, where $E(G)$ is the edge set of $G$, and
Ren qingcuo +3 more
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AKCE International Journal of Graphs and Combinatorics, 2018 Let G be a 4-connected graph, and let E ̃ (G) denote the set of those edges of G which are not contained in a triangle, and let E c (G) denote the set of 4-contractible edges of G . We show that if 3 ≤ | E ̃ (G) | ≤ 4 or | E ̃ (G) | ≥ 7 , then | E c (G) |
Yoshimi Egawa +2 more
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Families of Integral Cographs within a Triangular Array
Special Matrices, 2020 The determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product ...
Ching Hsin-Yun +2 more
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Sufficient conditions for triangle-free graphs to be super-$λ'$ [PDF]
Transactions on Combinatorics, 2018 An edge-cut $F$ of a connected graph $G$ is called a restricted edge-cut if $G-F$ contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity $λ'(G)$ of $G$. A graph $G$ is said to
Huiwen Cheng, Yan-Jing Li
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Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable [PDF]
Discrete Mathematics & Theoretical Computer Science, 2016 For planar graphs, we consider the problems of list edge coloring and list total coloring. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors.
Marthe Bonamy +2 more
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Decompositions of triangle-dense graphs [PDF]
Proceedings of the 5th conference on Innovations in theoretical computer science, 2014 High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense ...
Rishi Gupta +2 more
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On hamiltonicity of 1-tough triangle-free graphs
Electronic Journal of Graph Theory and Applications, 2021 Let ω(G) denote the number of components of a graph G. A connected graph G is said to be 1-tough if ω(G − X)≤|X| for all X ⊆ V(G) with ω(G − X)>1. It is well-known that every hamiltonian graph is 1-tough, but that the reverse statement is not true in ...
Wei Zheng, Hajo Broersma, Ligong Wang
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