Automorphisms and regular embeddings of merged Johnson graphs

Automorphisms of infinite Johnson graphs

Let I be a set of infinite cardinality α. For every cardinality β ≤ α the Johnson graphs Jβ and J are the graphs whose vertices are subsets X ⊂ I satisfying |X | = β , |I \ X | = α and |X | = α, |I \ X | = β (respectively) and vertices X, Y are adjacent if |X \ Y | = |Y \ X | = 1. Note that Jα = J and Jβ is isomorphic to J for every β < α. If β is finite then Jβ and J are connected and it is no...

Isometric embeddings of Johnson graphs in Grassmann graphs

Let V be an n-dimensional vector space (4 ≤ n < ∞) and let Gk(V ) be the Grassmannian formed by all k-dimensional subspaces of V . The corresponding Grassmann graph will be denoted by Γk(V ). We describe all isometric embeddings of Johnson graphs J (l,m), 1 < m < l − 1 in Γk(V ), 1 < k < n − 1 (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of J (n, k...

Labeling Subgraph Embeddings and Cordiality of Graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

On Automorphisms of Strongly Regular Graphs

A strongly regular graph with λ = 0 and μ = 3 is of degree 3 or 21. The automorphisms of prime order and the subgraphs of their fixed points are described for a strongly regular graph Γ with parameters (162, 21, 0, 3). In particular, the inequality |G/O(G)| ≤ 2 holds true for G = Aut(Γ).

Strongly regular graphs with non-trivial automorphisms