Related papers: A characterization of the Grassmann graphs: one mi…

We discuss the Grassmann graph $J_q(N,D)$ with $N \geq 2D$, having as vertices the $D$-dimensional subspaces of an $N$-dimensional vector space over the finite field $\mathbb{F}_q$. This graph is distance-regular with diameter $D$; to avoid…

In this paper we present a characterisation, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn).

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…

We prove that the number s(n) of disjoint minimal graphs supported on domains in R^n is bounded by e(n+1)^2. In the two-dimensional case we show that s(2) is at most three (the conjectured number is two).

The interval number of a graph $G$ is the minimum $k$ such that one can assign to each vertex of $G$ a union of $k$ intervals on the real line, such that $G$ is the intersection graph of these sets, i.e., two vertices are adjacent in $G$ if…

We compute the automorphism scheme of a generic odd dimensional $(2,2)$-complete intersection in characteristic $2$. This is the only case for complete intersections having a non-trivial identity component in automorphism schemes apart from…

We give a characterization of distance--preserving subgraphs of Johnson graphs, i.e. of graphs which are isometrically embeddable into Johnson graphs (the Johnson graph $J(m,\Lambda)$ has the subsets of cardinality $m$ of a set $\Lambda$ as…

We prove that an eulerian graph $G$ admits a decomposition into $k$ closed trails of odd length if and only if and it contains at least $k$ pairwise edge-disjoint odd circuits and $k\equiv |E(G)|\pmod{2}$. We conjecture that a connected…

A path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number $p(G)$ of graph $G$ is the cardinality of a path cover with the minimum number of paths. Reed in…

Let $G$ denote a $Q$-polynomial distance-regular graph with diameter $D$ at least 4. Assume that the intersection numbers of $G$ satisfy $a_i=0$ for $0 \leq i \leq D-1$ and $a_D\neq 0$. We show that $G$ is a polygon, a folded cube, or an…

Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with path-width at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight…

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection…

A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…

Let $G$ be a simple connected graph with $n\geq 5$ vertices. In this note, we will prove that $s_3(G)\leq n$, and characterize the graphs which satisfy that $s_3(G)=n, n-1, n-2, $ or $n-3$, where $s_3(G)$ is the third invariant factor of…

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…

In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy…

Let $\mathcal{G}(n,k)$ be the set of connected graphs of order $n$ with one of the Laplacian eigenvalue having multiplicity $k$. It is well known that $\mathcal{G}(n,n-1)=\{K_n\}$. The graphs of $\mathcal{G}(n,n-2)$ are determined by Das,…

We consider a distance-regular graph $\G$ with diameter $d \ge 3$ and eigenvalues $k=\theta_0>\theta_1>... >\theta_d$. We show the intersection numbers $a_1, b_1$ satisfy $$ (\theta_1 + {k \over a_1+1}) (\theta_d + {k \over a_1+1}) \ge -…

We give a complete characterization of mixed unit interval graphs, the intersection graphs of closed, open, and half-open unit intervals of the real line. This is a proper superclass of the well known unit interval graphs. Our result solves…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…