Judicious partition problems ask for partitions of the vertex set of graphs so that several quantities are optimized simultaneously. In this paper, we answer the following judicious partition question of Bollobás and Scott [6] in the affirmative: For any positive integer k and for any graph G of size m, does there exist a partition of V (G) into V1, . . . , Vk such that the total number of edge...

We extend the quasi-tree expansion of A. Champanerkar, I. Kofman, and N. Stoltzfus to not necessarily orientable ribbon graphs. We study the duality properties of the Bollobás-Riordan polynomial in terms of this expansion. As a corollary, we get a “connected state” expansion of the Kauffman bracket of virtual link diagrams. Our proofs use extensively the partial duality of S. Chmutov.

We introduce a family of number triangles defined by exponential Riordan arrays, which generalize Pascal’s triangle. We characterize the row sums and central coefficients of these triangles, and define and study a set of generalized Catalan numbers. We establish links to the Hermite, Laguerre and Bessel polynomials, as well as links to the Narayana and Lah numbers.

In this note we adapt a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125–148] from random graphs to random r-uniform hypergaphs. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs.

Let d ≥ 2 be given and let μ be an involution-invariant probability measure on the space of trees T ∈ Td with maximum degrees at most d. Then μ arises as the local limit of some sequence {Gn} ∞ n=1 of graphs with all degrees at most d. This answers Question 6.8 of Bollobás and Riordan [4].

In [4], the authors studied the Pascal matrix and the Stirling matrices of the first kind and the second kind via the Fibonacci matrix. In this paper, we consider generalizations of Pascal matrix, Fibonacci matrix and Pell matrix. And, by using Riordan method, we have factorizations of them. We, also, consider some combinatorial identities.

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.

Recently, in the paper (Cheon et al. Linear Algebra Appl 579:89–135, 2019) we suggested two conjectures about diameter of io-decomposable Riordan graphs Bell type. In this paper, give a counterexample for first conjecture. Then prove that conjecture is true some particular size and propose new Finally, show second special graphs.

INTRODUCTION • Human communication increasingly relies on technology • Text based communication (TBC) lacks communicative richness, as compared to face to communication (Harris & Paradice, 2007) • However, typographic cues may add informativeness by mimicking tone of voice in TBC (Riordan & Kreuz, 2010) • Gunraj et al. (2016) found one word response with ending period is perceived as rude/ insi...