The National University of San Luis (in Castilian, Universidad Nacional de San Luis, UNSL) is a public university in Argentina, with its seat in the city of San Luis, capital of the province of the same name, in the Cuyo region. It was created in 1973, along with the National University of San Juan, split off the National University of Cuyo based in Mendoza..
The rotation graph ℛ(G) is the graph whose vertices correspond to search trees on a graph G, with edges determined by rotation operations. In this paper, we analyze how the structure of a rotation graph changes when certain operations are applied to the underlying graph G. Specifically, we examine the effects of three key operations: adding a pendant vertex, adding a true twin to a vertex, and adding a false twin to a vertex. For each of these operations, we provide a full structural characterization of the new rotation graph. Using these descriptions, we investigate the chromatic number of rotation graphs, identifying conditions under which this parameter remains unchanged. As an application, we show that the chromatic number of the rotation graphs of non-complete threshold graphs (including complete split graphs and star graphs) and complete bipartite graphs is 3.
In this paper, we study a higher‐order Laplacian operator in the framework of Orlicz–Sobolev spaces, the biharmonic g ‐Laplacian where , with being an N ‐function. This operator is a generalization of the so‐called bi‐harmonic Laplacian . Here, we also establish basic functional properties of , which can be applied to existence results. Afterwards, we study the eigenvalues of , which depend on normalization conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to and we show regimes where the corresponding spectrum concentrates at 0, or coincide with .
A graph is said to be a Sterboul–Deming graph if KE(G)=∅, that is, if every vertex of G belongs to a posy or a flower (structures introduced by Sterboul, Deming, and Edmonds). These graphs can be regarded as the structural counterparts of König–Egerváry graphs. In this paper, we present several characterizations of Sterboul–Deming graphs. We first study the case of graphs with a perfect matching and with a unique perfect matching, providing a constructive algorithm to obtain the decomposition (SD(G), KE(G)). Then, we extend the analysis to the general case through the Gallai–Edmonds decomposition. In addition, we show that the class of Sterboul–Deming graphs is remarkably broad: it contains all graphs having a {C_n : n odd}-factor, providing a simple structural criterion for identifying such graphs. These results establish new connections between classical decomposition theorems and the internal structure of non–König–Egerváry graphs.
We study the nullspace of the adjacency matrix of split graphs, whose vertex set can be partitioned into a clique and an independent set. We introduce the clique-kernel, a subspace that decides whether clique vertices lie in the support of a kernel eigenvector, and we prove that its dimension is at most one. This yields the formula nl(Sp) = nl(R) + dim(Cker (Sp)), which fully describes the nullity of a split graph in terms of its biadjacency submatrix R. We also analyze unbalanced split graphs through the concept of swing vertices and characterize the structure of their kernel supports. Furthermore, we study the behavior of the nullspace under Tyshkevich composition and derive a closed formula for the determinant. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
An almost bipartite graph is a graph with a unique odd cycle. Levit and Mandrescu showed that in every non-König–Egerváry almost bipartite graph the equalities ker(G)=core(G), corona(G)∪ N(core(G)) = V(G) and |corona(G)|+|core(G)|=2α(G)+1 hold. In this work, we present a generalization of this theory by introducing the family of R-disjoint graphs, which contains all non-König–Egerváry almost bipartite graphs, allowing the presence of multiple odd cycles under connectivity constraints based on the reach sets R(C). We prove that R-disjoint graphs preserve the fundamental properties of almost bipartite graphs: ker(G)=core(G) and corona(G)∪ N(core(G))=V(G). Moreover, we establish the formula |corona(G)|+|core(G)|=2α(G)+k, where k is the number of disjoint odd cycles in G, which refines the previously known particular case when k=1. R-disjoint graphs naturally induce a canonical decomposition; we obtain structural properties of this decomposition and, as a consequence, verify a recent conjecture of Levit and Mandrescu.
