The relative fixity of a permutation group is the maximum proportion points fixed by non-trivial element group, and graph its automorphism viewed as on vertex-set graph. We prove in this paper that connected 2-arc-transitive graphs valence tends to 0 number vertices grows infinity. same result for class arc-transitive prime valence, more generally, any locally-L graphs, where L quasiprimitive g...

In this paper, a complete classification of finite simple cubic vertex-transitive graphs girth 6 is obtained. It proved that every such graph, with the exception Desargues graph on 20 vertices, either skeleton hexagonal tiling torus, truncation an arc-transitive triangulation closed hyperbolic surface, or 6-regular respect to dihedral scheme. Cubic larger than are also discussed.

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane as vertices and an adjacency resembling that of star Cayley graphs. The graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copi...

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph ΓB relative to B, the bipartite subgraph Γ[B,C] of Γ induced by adjacent blocks B,C of ΓB and a certain 1-design D(B) induced by a block B ∈ B. The present...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

This paper develops methods to reason about graph transformation rules for proving the preservation of structural properties, especially global properties on reachability. We characterize a graph transformation rule with an applicability condition specifying the matching conditions of the rule on a host graph as well as the properties to be preserved during the transformation. Our previous work...

Proof. Consider a graph like the one below, where there are three rows of n vertices. Given two boolean matrices A and B, we can create such a graph by adding an edge between the i vertex of the first row and the j of the second row iff Aij == 1. Construct edges between the second and third rows in a similar fashion for B. Thus, the product AB can be computed by taking the transitive closure of...

The spined cube SQn, as a variant network of the hypercube Qn, was proposed in 2011 and has attracted much attention because its smaller diameter. It is well-known that Qn Cayley graph. In present paper, we show SQn an m-Cayley graph, automorphism group semiregular subgroup acting on vertices with m orbits, where m=4 when n≥6 m=⌊n/2⌋ n≤5. Consequently, it shows can be partitioned into eight dis...

The (k, d)-hypersimplex is a (d − 1)-dimensional polytope whose vertices are the (0, 1)vectors that sum to k. When k = 1, we get a simplex whose graph is the complete graph Kd. Here we show how many of the well known graph parameters and attributes of Kd extend to a more general case. In particular we obtain explicit formulas in terms of d and k for the number of vertices, vertex degree, number...

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...