Related papers: Bonnet-Myers sharp graphs of diameter three
We look at the question of which distance-regular graphs are core-complete, meaning they are isomorphic to their own core or have a complete core. We build on Roberson's homomorphism matrix approach by which method he proved the…
This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $\c{D}(M,N)$ is finite for any…
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…
A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gr\"{o}bner basis with respect to the lexicographic order induced by $x_1 > \cdots > x_n > y_1> \cdots > y_n$. In this…
In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs…
A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more…
Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no…
Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional…
The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity…
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have…
We introduce and study a strong "thin triangle"' condition for directed graphs, which generalises the usual notion of hyperbolicity for a metric space. We prove that finitely generated left cancellative monoids whose right Cayley graphs…
A graph algorithm is truly subquadratic if it runs in ${\cal O}(m^b)$ time on connected $m$-edge graphs, for some positive $b < 2$. Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no…
Motivated by the theory of crystallizations, we consider an equivalence relation on the class of $3$-regular colored graphs and prove that up to this equivalence (a) there exists a unique contracted 3-regular colored graph if the number of…
A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on $n$ vertices is at most $\lfloor…
We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map…
In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-\'Emery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation…
We find a sharp bound for the order of the automorphism group of a stable curve of genus $g$ with $3g-3$ nodes, and a sharp bound for the order of the automorphism group of such a curve with all smooth components. Combined with the results…
An n-simplex is said to be n-well-centered if its circumcenter lies in its interior. We introduce several other geometric conditions and an algebraic condition that can be used to determine whether a simplex is n-well-centered. These…
We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the M\"obius…
The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdoes-Renyi…