1. Uniform equations for bipartite graphs and the center of a Terwilliger algebraŠtefko Miklavič, Giusy Monzillo, 2026, original scientific article Abstract: The uniform property was introduced by P. Terwilliger in the context of graded posets and was later extended to connected bipartite graphs. The core of this definition involves the so called uniform equations that must be satisfied. Let Γ denote a connected bipartite graph. Fix a vertex x of Γand let T=T(x) denote the corresponding Terwilliger algebra. In this paper, we study the connections between the uniform equations and the center of T. We show that these uniform equations give rise to a certain subspace of the center of T. Changing the logical direction, we show that if a matrix of a particular form belongs to the center of T, then uniform equations are satisfified.
Keywords: uniform equations, center of a Terwilliger algebra, bipartite graphs
Published in RUP: 08.05.2026; Views: 387; Downloads: 9
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2. Automorphisms and quotients of 2-colored quasi best match graphsAnnachiara Korchmaros, 2026, original scientific article Abstract: 2-colored quasi best match graphs (2-qBMGs) are directed graphs that arose in evolution theory. Investigations of 2-qBMGs have mostly focused on computational issues. However, 2-qBMGs also have relevant properties for structural graph theory; in particular, their undirected underlying graph is free from induced paths and cycles of size at least 6. In this paper, results on the structure of the automorphism groups of 2-qBMGs are obtained, which shows how to construct 2-qBMGs with large automorphism groups.
Keywords: group of automorphisms, bipartite graphs, phylogenetics
Published in RUP: 05.01.2026; Views: 1390; Downloads: 3
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6. On the connectivity of bipartite distance-balanced graphsŠtefko Miklavič, Primož Šparl, 2012, original scientific article Abstract: A connected graph ▫$\varGamma$▫ is said to be distance-balanced whenever for any pair of adjacent vertices ▫$u,v$▫ of ▫$\varGamma$▫ the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. In [K. Handa, Bipartite graphs with balanced ▫$(a,b)$▫-partitions, Ars Combin. 51 (1999), 113-119] Handa asked whether every bipartite distance-balanced graph, that is not a cycle, is 3-connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distance-balanced graph, that is not a cycle and is not 3-connected, has 18 vertices and is unique. In addition, we give a complete classification of non-3-connected bipartite distance-balanced graphs for which the minimal distance between two vertices in a 2-cut is three. All such graphs are regular and for each ▫$k \geq 3$▫ there exists an infinite family of such graphs which are ▫$k$▫-regular.Furthermore, we determine a number of structural properties that a bipartite distance-balanced graph, which is not 3-connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distance-balanced graphs.
Keywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Published in RUP: 15.10.2013; Views: 10770; Downloads: 105
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