On (4,2)-digraph Containing a Cycle of Length 2

A diregular digraph is a digraph with the in-degree and out-degree of all vertices is constant. The Moore bound for a diregular digraph of degree d and diameter k is M_{d,k}=l+d+d^2+...+d^k. It is well known that diregular digraphs of order M_{d,k}, degree d>l tnd diameter k>l do not exist . A (d,k) -digraph is a diregular digraph of degree d>1, diameter k>1, and number of vertices one less than the Moore bound. For degrees d=2 and 3,it has been shown that for diameter k >= 3 there are no such (d,k)-digraphs. However for diameter 2, it is known that (d,2)-digraphs do exist for any degree d. The line digraph of K_{d+1} is one example of such (42)-digraphs. Furthermore, the recent study showed that there are three non-isomorphic(2,2)-digraphs and exactly one non-isomorphic (3,2)-digraph. In this paper, we shall study (4,2)-digraphs. We show that if (4,2)-digraph G contains a cycle of length 2 then G must be the line digraph of a complete digraph K_5

Non-singular circulant graphs and digraphs

We give necessary and sufficient conditions for a few classes of known circulant graphs and/or digraphs to be singular. The above graph classes are generalized to $(r,s,t)$-digraphs for non-negative integers $r,s$ and $t$, and the digraph $C_n^{i,j,k,l}$, with certain restrictions. We also obtain a necessary and sufficient condition for the digraphs $C_n^{i,j,k,l}$ to be singular. Some necessary conditions are given under which the $(r,s,t)$-digraphs are singular.Comment: 12 page

About the existence of oriented paths with three blocks

A path P(k,l,r) is an oriented path consisting of k forward arcs, followed by l backward arcs, and then by r forward arcs. We prove the existence of any oriented path of length n-1 with three blocks having the middle block of length one in any (2n-3)- chromatic digraph, which is an improvement of the latest bound reached in this case. Concerning the general case of paths with three blocks, we prove, after partitioning the problem into three cases according to the value of k,l and r that the chromatic number of digraphs containing no P(k,l,r) of length n-1 is bounded above by 2(n-1)+r, 2(n-1)+l+r-k and 2(n+l-1)-k in the three cases respectively

On hardware for generating routes in Kautz digraphs

In this paper we present a hardware implementation of an algorithm for generating node disjoint routes in a Kautz network. Kautz networks are based on a family of digraphs described by W.H. Kautz[Kautz 68]. A Kautz network with in-degree and out-degree d has N = dk + dk¿1 nodes (for any cardinals d, k>0). The diameter is at most k, the degree is fixed and independent of the network size. Moreover, it is fault-tolerant, the connectivity is d and the mapping of standard computation graphs such as a linear array, a ring and a tree on a Kautz network is straightforward.\ud The network has a simple routing mechanism, even when nodes or links are faulty. Imase et al. [Imase 86] showed the existence of d node disjoint paths between any pair of vertices. In Smit et al. [Smit 91] an algorithm is described that generates d node disjoint routes between two arbitrary nodes in the network. In this paper we present a simple and fast hardware implementation of this algorithm. It can be realized with standard components (Field Programmable Gate Arrays)

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.Comment: 17 pages, 3 figure

Non-commutative counting invariants and curve complexes

In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of non-commutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The non-commutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition defines much larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on examples and extend our studies beyond counting. We enrich our invariants with structures: the inclusion of subcategories makes them partially ordered sets, and considering semi-orthogonal pairs of subcategories as edges amount to directed graphs. In addition to computing the non-commutative curve-counting invariants in $D^b(Q)$ for two affine quivers, for $A_n$ and $D_4$ we derive formulas for counting of the subcategories of type $D^b(A_k)$ in $D^b(A_n)$, whereas for the two affine quivers and for $D_4$ we determine and count all generated by an exceptional collection subcategories. Estimating the numbers counting non-commutative curves in $D^b({\mathbb P}^2)$ modulo group action we prove finiteness and that an exact determining of these numbers leads to proving (or disproving) of Markov conjecture. Regarding the mentioned structure of a partially ordered set we initiate intersection theory of non-commutative curves. Via the structure of a directed graph we build an analogue to the classical curve complex used in Teichmueller and Thurston theory. The paper contains many pictures of graphs and presents an approach to Markov Conjecture via counting of subgraphs in a graph associated with $D^b(P^2)$. Some of the results proved here were announced in the previous work.Comment: In v4, 65 pages, we have reorganized the paper and removed some inaccuracies. Sections 2 to 7 are dedicated to general theory and then follow sections with examples. In the previous version the letter $\mathcal J$ in the definition of $C_{\mathcal J, P}(\mathcal T)$ was a set of non-trivial pairwise non-equivalent triangulated categories. Now we remove the restriction of non-trivialit