An alternate description of a (q + 1; 8)-cage

Let q >= 2 be a prime power. In this note we present an alternate description of the known (q + 1; 8)-cages which has allowed us to construct small (k; g)–graphs for k =q, q-1, and g = 7; 8 in other papers on this same topic.Peer ReviewedPostprint (published version

A construction of Small (q-1)-Regular Graphs of Girth 8

In this note we construct a new infinite family of (q - 1)-regular graphs of girth 8 and order 2q(q - 1)(2) for all prime powers q >= 16, which are the smallest known so far whenever q - 1 is not a prime power or a prime power plus one itself.Peer ReviewedPostprint (author’s final draft

Small regular graphs of girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain (q + 1)-regular graphs of girth 7 and order 2q(3) + q(2) + 2q for each even prime power q >= 4, and of order 2q(3) + 2q(2) q + 1 for each odd prime power q >= 5.Postprint (published version

On generating Cayley's Graphs

In this paper, we obtain some infinite families of graphs appearing as Cayley graphs in $\mathbb{Z}_n$ and $D_n$. Furthermore, a list of all the Cayley graphs coming from the groups up to order twelve is presented

2-factors of regular graphs: a survey

A 2–factor of a graph G is a 2–regular spanning subgraph of G. We survey results on the structure of 2–factors in regular graphs obtained in the last years by several authors

k-path connectivity and mk-generation, an upper bound

We consider simple connected graphs for which there is a path of length at least k between every pair of distinct vertices. We wish to show that in these graphs the cycle space over $\mathbb{Z}_2$ is generated by the cycles of length at least $mk$, where $m = 1$ for $3 \le k \le 6$, $m = 6/7$ for $k = 7$, $m \ge 1/2$ for $k \ge 8$ and $m \le 3/4 + 0(1)$ for large k

Nonseparating n-trees of diameter at most 4 in (2n+2)-cohesive graphs.

A connected simple graph G is called $k$--cohesive if for any pair of distinct vertices $u,v \in V(G)$, $d(u) + d(v) + d(u,v) \ge k$. A subgraph $H$ of a connected graph $G$ is non-separating if $G -V(H)$ is connected. Locke [MAA Monthly - 1998] conjectured that given a tree $T$ on $n$ vertices, $n \ge 3$, any $2n$--cohesive graph has a non-separating copy of $T$. Here we prove that given a tree $T$ on $n$ vertices and diameter at most 4, any $(2n + 2)$--cohesive graph has a non-separating copy of $T$

Adjacency matrices of polarity graphs and other C_4-free graphs of large size

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph Gq from a projective plane $PG(2, q)$, where $q$ is a prime power. Denote by $ex(n;C_4)$ the maximum number of edges of a graph on $n$ vertices and free of squares $C_4$. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function $ex(n;C_4)$, for some $n < q^2 + q + 1$. In particular, we construct a $C_4$--free graph on $n = q^2 − \sqrt{q}$ vertices and $\frac{1}{2}q(q^2 − 1) − \frac{1}{2} \sqrt{q}(q − 1)$ edges, for a square prime power $q$

Pseudo and Strongly Pseudo 2-Factor Isomorphic Regular Graphs and Digraphs

A graph G is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of G. In Abreu et al. (2008) [3] we proved that pseudo 2-factor isomorphic k-regular bipartite graphs exist only for k ≤ 3. In this paper we generalize this result for regular graphs which are not necessarily bipartite. We also introduce strongly pseudo 2-factor isomorphic graphs and we prove that pseudo and strongly pseudo 2-factor isomorphic 2k-regular graphs and k-regular digraphs do not exist for k ≥ 4. Moreover, we present constructions of infinite families of regular graphs in these classes. In particular we show that the family of Flower snarks is strongly pseudo 2-factor isomorphic but not 2-factor isomorphic and we conjecture that, together with the Petersen and the Blanuša2 graphs, they are the only cyclically 4-edge-connected snarks for which each 2-factor contains only cycles of odd length