On tetravalent half-arc-transitive graphs of girth 5

A subgroup of the automorphism group of a graph \G is said to be {\em half-arc-transitive} on \G if its action on \G is transitive on the vertex set of \G and on the edge set of \G but not on the arc set of \G. Tetravalent graphs of girths $3$ and $4$ admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth $5$. We show that, with two exceptions, all such graphs only have directed $5$-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed $5$-cycles, study some of their graph theoretic properties and prove that the $5$-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth $5$ admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth $5$

Distance-unbalancedness of graphs

In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of well-known graphs. Distance-unbalancedness of trees is also studied. A few conjectures are stated and some open problems are proposed.Comment: 14 pages, 3 figure

Strongly distance-balanced graphs and graph products

AbstractA graph G is strongly distance-balanced if for every edge uv of G and every i≥0 the number of vertices x with d(x,u)=d(x,v)−1=i equals the number of vertices y with d(y,v)=d(y,u)−1=i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distance-balanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given

Almost all quartic half-arc-transitive weak metacirculants of Class II are of Class IV

AbstractA half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. A weak metacirculant is a graph admitting a transitive metacyclic group that is a group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1 and n≥2, and where σ normalizes ρ. It was shown in [D. Marušič, P. Šparl, On quartic half-arc-transitive metacirculants, J. Algebr. Comb. 28 (2008) 365–395] that each connected quartic half-arc-transitive weak metacirculant X belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph Xρ relative to the semiregular automorphism ρ. The first of these classes, called Class I, coincides with the class of so-called tightly attached graphs. Class II consists of the quartic half-arc-transitive weak metacirculants for which the quotient graph Xρ is a cycle with a loop at each vertex. Class III consists of those graphs for which each vertex of the quotient graph Xρ is connected to three other vertices, to one with a double edge. Finally, Class IV consists of those graphs for which Xρ is a simple quartic graph.This paper consists of two results concerning graphs of Class II. It is shown that, with the exception of the Doyle–Holt graph and its canonical double cover, each quartic half-arc-transitive weak metacirculant of Class II is also of Class IV. It is also shown that although quartic half-arc-transitive weak metacirculants of Class II which are not tightly attached exist they are “almost tightly attached”. More precisely, their radius is at most four times their attachment number

GeoGebrina orodja za simbolno računanje

Pred vami je delovno gradivo z zgoščeno predstavitvijo vsebine de- lavnic, ki smo jih pripravili dr. Marko Razpet, dr. Marko Slapar, dr. Primož Šparl in podpisani. Z na- slednjimi izvedbami seminarja bomo gradivo še nekoliko dopolnili in izboljšali, upam pa, da vam bo že ta različica v pomoč pri vašem samostojnem delu. Gradivo je bilo pripravljeno za verzijo GeoGebra 4.2 (november 2012), ki je prosto dostopna na spletnem naslovu www.geogebra.org. Priročnik za uporabo GeoGebre v angleškem jeziku je na voljo na naslovu http://www.geogebra.org/book/intro-en.pdf

Dosegljivostne relacije, tranzitivni digrafi in grupe

In [A. Malnič, D. Marušič, N. Seifter, P. Šparl and B. Zgrablič, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566-1581] it was shown that properties of digraphs such as growth, property ▫$mathbf{Z}$▫, and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs. In this paper we study these relations in connection with certain properties of automorphism groups of transitive digraphs. In particular, one of the main results shows that if atransitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, then its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question. The obtained results have interesting implications for Cayley digraphs of certain types of groups such as torsion-free groups of polynomial growth.V [A. Malnič, D. Marušič, N. Seifter, P. Šparl and B. Zgrablič, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566-1581] je bilo pokazano, da se lastnosti usmerjenih grafov, kot so rast, lastnost ▫$mathbf{Z}$▫, in število koncev odražajo v lastnostih določenih dosegljivostnih relacij definiranih na vozliščih ustreznih digrafov. V tem članku obravnavamo te relacije v povezavi z določenimi lastnostmi grup avtomorfizmov tranzitivnih digrafov. Posebej, eden od glavnih rezultatov kaže, da če tranzitivni digraf dopušča nilpotentno podgrupo avtomorfizmov s končno mnogo orbitami, potem sta njegov nilpotentni razred in število obit tesno povezana z določenimi lastnostmi dosegljivostnih relacij, definiranih na ustreznih digrafih. Dobljeni rezultati imajo zanimive implikacije za Cayleyeve digrafe določenih tipov grup, kot so grupe brez torzije s polinomsko rastjo