Random Cyclic Triangle-Free Graphs of Prime Order

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

Empirical study prove that breadth-first search is more effective memory usage than depth-first search in frontier boundary cyclic graph

<p><span id="docs-internal-guid-06e4528a-7fff-0e38-150e-f136d6f22d84"><span>Memory consumption, of opened and closed lists in graph searching algorithms, affect in finding the solution. Using frontier boundary will reduce the memory usage for a closed list, and improve graph size expansion. The blind algorithms, depth-first frontier Searches, and breadth-first frontier Searches were used to compare the memory usage in slide tile puzzles as an example of the cyclic graph. This paper aims to prove that breadth-first frontier search is better than depth-first frontier search in memory usage. Both opened and closed lists in the cyclic graph are used. The level number and nodes count at each level for slide tile puzzles are changed when starting from different empty tile location. Eventually, the unorganized spiral path in depth-first search appears clearly through moving inside the graph to find goals.</span></span></p>

Two-Distance-Primitive Graphs

A 2-distance-primitive graph is a vertex-transitive graph whose vertex stabilizer is primitive on both the first step and the second step neighborhoods. Let $\Gamma$ be such a graph. This paper shows that either $\Gamma$ is a cyclic graph, or $\Gamma$ is a complete bipartite graph, or $\Gamma$ has girth at most $4$ and the vertex stabilizer acts faithfully on both the first step and the second step neighborhoods. Also a complete classification is given of such graphs satisfying that the vertex stabilizer acts $2$-transitively on the second step neighborhood. Finally, we determine the unique 2-distance-primitive graph which is locally cyclic.

THE CYCLIC GRAPH OF A Z-GROUP

For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.

Characteristic Graph vs Benzenoid Graph

A characteristic graph is a tree representative of its corresponding benzenoid (cyclic) graph. It may contain necessary information of several properties of benzenoids. The PI-index of benzenoids and their characteristic graphs are compared by correlating it to a structural property (π-electron energy) of the benzenoids using MLR analysis. PI index being applicable to both trees and cyclic graphs, yielded required results for benzenoid and their characteristic graph.

On the non-cyclic graph of a group

Here we study some algebraic properties of non-cyclic graphs. In this paper, we show that [Formula: see text] is isomorphic to [Formula: see text] or [Formula: see text] if and only if [Formula: see text] is isomorphic to [Formula: see text] or [Formula: see text], respectively. We characterize all groups of order [Formula: see text] like [Formula: see text] in which [Formula: see text], where [Formula: see text].