A formulation of a (q+1,8)-cage

Let $q\ge 2$ be a prime power. In this note we present a formulation for obtaining the known $(q+1,8)$-cages which has allowed us to construct small $(k,g)$--graphs for $k=q-1, q$ and $g=7,8$. Furthermore, we also obtain smaller $(q,8)$-graphs for even prime power $q$.Comment: 14 pages, 2 figure

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 $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.Comment: 8 pages, 2 figure

A new bound for the connectivity of cages

AbstractAn (r,g)-cage is an r-regular graph of girth g of minimum order. We prove that all (r,g)-cages are at least ⌈r/2⌉-connected for every odd girth g≥7 by means of a matrix technique which allows us to construct graphs without short cycles. This lower bound on the vertex connectivity of cages is a new advance in proving the conjecture of Fu, Huang and Rodger which states that all (r,g)-cages are r-connected

On the Minimum Order of Extremal Graphs to have a Prescribed Girth

We show that any n‐vertex extremal graph G without cycles of length at most k has girth exactly $k+1$ if $k\ge 6$ and $n>(2(k-2)^{k-2}+k-5)/(k-3)$. This result provides an improvement of the asymptotical known result by Lazebnik and Wang [J. Graph Theory, 26 (1997), pp. 147–153] who proved that the girth is exactly $k+1$ if $k\ge 12$ and $n\ge 2^{a^2+a+1}k^a$, where $a=k-3-\lfloor(k-2)/4\rfloor$. Moreover, we prove that the girth of G is at most $k+2$ if $n>(2(t-2)^{k-2}+t-5)/(t-3)$, where $t=\lceil (k+1)/2\rceil\ge 4$. In general, for $k\ge 5$ we show that the girth of G is at most $2k-4$ if $n\ge 2k-2$

An explicit formula for obtaining (q+1,8)(q+1,8)-cages and others small regular graphs of girth 8

Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The first contribution of this paper is a construction of these graphs in an alternative way by means of an explicit formula using graphical terminology. Furthermore by removing some specific perfect dominating sets from a $(q+1,8)$-cage we derive $k$-regular graphs of girth 8 for $k= q-1$ and $k=q$, having the smallest number of vertices known so far

Families of Small Regular Graphs of Girth 5

In this paper we obtain $(q+3)$--regular graphs of girth 5 with fewer vertices than previously known ones for $q=13,17,19$ and for any prime $q \ge 23$ performing operations of reductions and amalgams on the Levi graph $B_q$ of an elliptic semiplane of type ${\cal C}$. We also obtain a 13-regular graph of girth 5 on 236 vertices from $B_{11}$ using the same technique

Soybean as supplement of growing cattle on tropical pasture: effects on intake, digestibility and animal performance

An indoor trial was conducted with weaning heif­ers (experiment 1). Raw whole soybean (RSB), whole soybean heated with steam (SHSB) and extruded whole soybean (EXSB) were given as supplements at 0.7% of live weight (LW). One group was the control, with no supplement. All supplements increased total dry matter (DM) intake and did not affect hay intake. RSB was the most degradable crude protein (CP) source. Total digestive tract digestibility was not affected by treatments, except for apparent CP digestibility. Because total DM intake increased in supplemented animals, all supple­ments increased intake of digestible DM, organic matter (OM) and CP over the control. In another assay (experiment 2) a performance trial was conducted with forty weaned heifers. The same supplements used in experiment 1 were used. Heifers supplemented with EXSB gained the highest and had better apparent feed efficiency. Apparent feed efficiency was similar to those observed in protein supplementation, suggesting that some increase in DM intake from pasture may have occurred. RSB had the poorest apparent feed efficiency which agreed with the numerical lower OM intake observed in experiment 1. All supplemented heifers had higher gain, hip height change, thoracic perimeter and body condition score change than the control group