The Merrifield-Simmons index is one of the most popular topological indices in chemistry and in mathematical properties; there is a correlation between this index and boiling points. The MerrifieldSimmons index of a graph is defined as the total number of its independent sets, including the empty set. This paper proposed an edge grafting theorem operation, which is certain kind of edge moving b...

Ryjacek [Z. Ryjáček: On a closure concept in claw-free graphs. Journal of Combinatorial Theory Ser. B 70 (1997), 217-224] defined a powerful closure operation cl(G) on claw-free graphs G. Very recently, Ryjacek, Yoshimoto and the talker developed the closure operation cl (G) on claw-free graphs which preserves the (non)-existence of a 2-factor. In this talk, we introduce a closure operation cl(...

The branching operation D, defined by Propp, assigns to any directed graph G another directed graph D(G) whose vertices are the oriented rooted spanning trees of the original graph G. We characterize the directed graphs G for which the sequence δ(G) = (G,D(G), D(G), . . .) converges, meaning that it is eventually constant. As a corollary of the proof we get the following conjecture of Propp: fo...

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E) and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n > 1, the unitary Cayley graph Xn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a...

Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition ...

We show that if the number of directions not determined by a pointset W of AG(3, q), q = p, of size q is at least pq then every plane intersects W in 0 modulo p points and apply the result to ovoids of the generalised quadrangles T2(O) and T ∗ 2 (O).

Cameron gave an upper bound for the size of any s-distance family of permutations. We prove a modulo p version of Cameron’s result. In the proof we use the polynomial subspace method. As an application we describe here an explicit construction which produces for every integer m > 1 a graph on at least m 2 9 logm log logm vertices containing neither a clique of size m nor an independent set of s...

Recently, Grynkiewicz et al. [Israel J. Math. 193 (2013), 359–398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence a1x1+ · · ·+akxk ≡ b (mod n), where a1, . . . , ak, b, n (n ≥ 1) are arbitrary integers, has a solution 〈x1, . . . , xk〉 ∈ Z k n with all xi distinct modulo n. So, it would be an interesting pro...

Hyperplanes of the form xj = xi + c are called affinographic. For an affinographic hyperplane arrangement in Rn, such as the Shi arrangement, we study the function f(m) that counts integral points in [1,m]n that do not lie in any hyperplane of the arrangement. We show that f(m) is a piecewise polynomial function of positive integers m, composed of terms that appear gradually as m increases. Our...

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flowinvariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Let Γ be a permutation group acting on the set of cells. We define the group network GΓ, whose architecture is entirely determined by the group orbits of Γ. We prove ...