1,422 research outputs found
Extremal Graphs for Intersecting Triangles
AbstractIt is known that for a graph on n vertices [n2/4] + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex
The -index of graphs without intersecting triangles/quadrangles as a minor
The -matrix of a graph is the convex linear combination of
the adjacency matrix and the diagonal matrix of vertex degrees ,
i.e., , where . The -index of is the largest eigenvalue of .
Particularly, the matrix (resp. ) is exactly the
adjacency matrix (resp. signless Laplacian matrix) of . He, Li and Feng
[arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency
spectral radius among all graphs of sufficiently large order without
intersecting triangles and quadrangles as a minor, respectively. Motivated by
the above results of He, Li and Feng, in this paper we characterize the
extremal graphs with maximum -index among all graphs of sufficiently
large order without intersecting triangles and quadrangles as a minor for any
, respectively. As by-products, we determine the extremal graphs
with maximum signless Laplacian radius among all graphs of sufficiently large
order without intersecting triangles and quadrangles as a minor, respectively.Comment: 15 page
A density Corr\'adi-Hajnal Theorem
We find, for all sufficiently large and each , the maximum number of
edges in an -vertex graph which does not contain vertex-disjoint
triangles.
This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in
turn an extension of Mantel's Theorem. Our result can also be viewed as a
density version of the Corradi-Hajnal Theorem.Comment: 41 pages (including 11 pages of appendix), 4 figures, 2 table
Core congestion is inherent in hyperbolic networks
We investigate the impact the negative curvature has on the traffic
congestion in large-scale networks. We prove that every Gromov hyperbolic
network admits a core, thus answering in the positive a conjecture by
Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which
is based on the experimental observation by Narayan and Saniee, Physical Review
E, 84 (2011) that real-world networks with small hyperbolicity have a core
congestion. Namely, we prove that for every subset of vertices of a
-hyperbolic graph there exists a vertex of such that the
disk of radius centered at intercepts at least
one half of the total flow between all pairs of vertices of , where the flow
between two vertices is carried by geodesic (or quasi-geodesic)
-paths. A set intercepts the flow between two nodes and if
intersect every shortest path between and . Differently from what
was conjectured by Jonckheere et al., we show that is not (and cannot be)
the center of mass of but is a node close to the median of in the
so-called injective hull of . In case of non-uniform traffic between nodes
of (in this case, the unit flow exists only between certain pairs of nodes
of defined by a commodity graph ), we prove a primal-dual result showing
that for any the size of a -multi-core (i.e., the number
of disks of radius ) intercepting all pairs of is upper bounded by
the maximum number of pairwise -apart pairs of
An extension of Tur\'an's Theorem, uniqueness and stability
We determine the maximum number of edges of an -vertex graph with the
property that none of its -cliques intersects a fixed set .
For , the -partite Turan graph turns out to be the unique
extremal graph. For , there is a whole family of extremal graphs,
which we describe explicitly. In addition we provide corresponding stability
results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's
comments incorporate
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