相关论文: There are only finitely many distance-regular grap…

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is…

Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the binomial random graph with the same number of vertices as $G$ and with edge…

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with…

In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with…

This paper considers three imprimitive distance-regular graphs with 486 vertices and diameter 4: the Koolen--Riebeek graph (which is bipartite), the Soicher graph (which is antipodal), and the incidence graph of a symmetric transversal…

We prove that, if $\Gamma$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\Gamma$ of order at least $6$, or the number of vertices of $\Gamma$ is bounded above by an absolute…

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where…

An $\epsilon$-distance-uniform graph is one in which from every vertex, all but an $\epsilon$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in…

We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six…

A subset $D\subseteq V(G)$ is called a $k$-distance dominating set of $G$ if every vertex in $V(G)\setminus D$ is within distance $k$ from some vertex of $D$. The minimum cardinality among all $k$-distance dominating sets of $G$ is called…

We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding an edge…

A 2-nearly Platonic graph of type (k|d) is a k-regular planar graph with f faces, f-2 of which are of degree d and the remaining two are of degrees m_1;m_2, both different from d. Such a graph is called balanced if m_1=m_2. We show that all…

The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the…

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph…

Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Also it is known that K7 and the thirteen graphs obtained from K7 by rY moves are intrinsically knotted graphs with 21 edges. We prove that these…

In this paper we show that for each $n$, there exists a simple graph whose reliability polynomial has at least $n$ inflection points.

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so…

A weakly distance-regular digraph is thick if its attached scheme is regular. In this paper, we show that each commutative thick weakly distance-regular digraph has a thick weakly distance-regular subdigraph such that the corresponding…

In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $\Gamma$ with parameters (216, 40, 4, 8) and proved that it is the unique PSU(4,2)-invariant…

Juri\'{s}i\v{c} et al. conjectured that if a distance-regular graph $\Gamma$ with diameter $D$ at least three has a light tail, then one of the following holds: 1.$a_1 =0$; 2.$\Gamma$ is an antipodal cover of diameter three; 3.$\Gamma$ is…