On algebraic constructions of graphs without small cycles and commutative diagrams and their applications On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Vasyl Ustimenko and Urszula Polubiec-Romanczuk Institute of Mathematics Maria Curie-Skłodowska University in Lublin, Poland MODERN TRENDS IN ALGEBRAIC THEORY Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction Recall that a girth is the length of a minimal cycle in a simple graph. Studies of maximal size ex ( C 3 , C 4 , . . . , C 2 m , v ) of the simple graph on v vertices without cycles of length 3 , 4 , . . . , 2 m , i. e. graphs of girth > 2 m , form an important direction of Extremal Graph Theory. Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction As it follows from the famous Even Circuit Theorem by P. Erd˝ os’ we have inequality ex ( C 3 , C 4 , . . . , C 2 m , v ) ≤ cv 1 + 1 / n , where c is a certain constant. The bound is known to be sharp only for n = 4 , 6 , 10. The first general lower bounds of kind ex ( v , C 3 , C 4 , . . . C n ) = Ω( v 1 + c / n ) , where c is some constant < 1 / 2 were obtained in the 50th by Erd˝ os’ via studies of families of graphs of large girth , i.e. infinite families of simple regular graphs Γ i of degree k i and order v i such that g (Γ i ) ≥ c log k i v i , where c is the independent of i constant. Erd˝ os’ proved the existence of such a family with arbitrary large but bounded degree k i = k with c = 1 / 4 by his famous probabilistic method. Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction Only two explicit families of regular simple graphs of large girth with unbounded girth and arbitrarily large k are known: the family X ( p , q ) of Cayley graphs for PSL 2 ( p ) , where p and q are primes, had been defined by G. Margulis [12] and investigated by A. Lubotzky, Sarnak [13] and Phillips and the family of algebraic graphs CD ( n , q ) [14]. Graphs CD ( n , q ) appears as connected components of graphs D ( n , q ) defined via system of quadratic equations. The best known lower bound for d � = 2 , 3 , 5 had been deduced from the existence of above mentioned families of graphs ex ( v , C 3 , C 4 , . . . , C 2 d ) ≥ cv 1 + 2 / ( 3 d − 3 + e ) where e = 0 if d is odd, and e = 1 if d is even. Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction Accordig to the theorem of Alon and Boppana, large enough members of an infinite family of q -regular graphs satisfy the inequality λ ≥ 2 √ q − 1 − o ( 1 ) , where λ is the second largest eigenvalue in absolute value. Ramanujan graphs are q -regular graphs for which the inequality λ ≤ 2 √ q − 1 holds. We say that a family of regular graphs of bounded degree q of increasing order n has an expansion constant c , c > 0 if for each subset A of the vertex set X , | X | = n with | A | ≤ n / 2 the inequality | ∂ A | ≥ c | A | holds. The expansion constant of the family of q -regular graphs Γ i , i = 1 , 2 , . . . can be bounded by below via upper limit β = ( q − λ i ) / q , i → ∞ , where λ i is the second largest eigenvalue of family Γ i . Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction The spectral gap of the family of q -regular graphs is the difference of q and upper limit λ for the second largest eigenvalues λ i of Γ i . Graphs Γ i form a family of expanders if its spectral gap distinct from zero. It is clear that a family of Ramanujan graphs of bounded degree q has the best possible spectral gap q − λ . We say, that family of q -regular graphs Γ i is a family of almost Ramanujan graphs if its second largest eigenvalues are bounded � above by 2 ( q ) . The above mentioned family X ( p , q ) is a family of Ramanujan graphs. That is why we refer to them as Cayley - Ramanujan graphs. The family CD ( n , q ) is a family of almost Ramanujan graphs. It is known that if q ≥ 5 these graphs are not Ramanujan despite the fact that projective limit of CD ( n , q ) is a q -regular tree. The reason is that the eigenspace of resulting infinite tree is not a Hilbert space (topology is p -adic). Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction Recall, that family of regular graphs Γ i of degree k i and increasing order v i is a family of graphs of small world if diam (Γ i ) ≤ c log k i ( v i ) for some independent constant c , c > 0, where diam (Γ i ) is diameter of graph G i . The graphs X ( p . q ) form a unique known family of large girth which is a family of small world graphs at the same time. There is a conjecture known from 1995 that the family of graphs CD ( n , q ) for odd q is another example of such kind. Currently. it is proven that the diameter of CD ( n , q ) is bounded from above by polynomial function d ( n ) , which does not dependent from q . Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction Expanding properties of X ( p , q ) and D ( n , q ) can be used in Coding Theory (magnifiers, super concentrators, etc ). The absence of short cycles and high girth property of both families can be used for the construction of LDPC codes [15]. This class of error correcting codes is an important tool of security for satellite communications. The usage of CD ( n , q ) as Tanner graphs producing LDPC codes lead to better properties of corresponding codes in the comparison to usage of Cayley - Ramanujan graphs (see [16]). Both families X ( p , q ) and CD ( n , q ) consist of edge transitive graphs. Their expansion properties and the property to be graphs of a large girth hold also for random graphs, which have no automorphisms at all. To make better deterministic approximation of random graph we can look at regular expanding graphs of a large girth without edge transitive automorphism group. Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction We consider below optimisation problem for simple graphs which is similar to a problem of finding maximal size for a graph on v vertices with the girth ≥ d . Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction Let us refer to the minimal length of a cycle, through the vertex of given vertex of the simple graph Γ as cycle indicator of the vertex . The cycle indicator of the graph Cind (Γ) will be defined as the maximal cycle indicator of its vertices. Regular graph will be called cycle irregular graph if its indicator differs from the girth (the length of the minimal cycle). The solution of the optimization problem of computation of the maximal size e = e ( v , d ) of the graph of order v with the size greater than d , d > 2 has been found very recently. It turns out that e ( v , d ) ⇔ O ( v 1 +[ 2 / d ] ) and this bound is always sharp (see [17] or [18] and further references). Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
On algebraic constructions of graphs without small cycles and commutative diagrams and their applications Introduction We refer to the family of regular simple graphs Γ i of degree k i and order v i as the family of graphs of large cycle indicator , if Cind (Γ i ) ≥ c log k i ( v i ) for some independent constant c , c > 0. We refer to the maximal value of c satisfying the above inequality as speed of growth of the cycle indicator for the family of graphs Γ i . As it follows from the written above evaluation of e ( v , d ) the speed of growth of the cycle indicator for the family of graphs of constant but arbitrarily large degree is bounded above by 2. Vasyl Ustimenko and Urszula Polubiec-Romanczuk On algebraic constructions of graphs without small cycles and
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