Laplacian integral graphs with a given degree sequence constraint

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Laplacian integral graphs with a given degreee sequence constraint

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) − A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral is all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all L-integral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Identification and characterization of constrained non-exonic bases lacking predictive epigenomic and transcription factor binding annotations

Annotations of evolutionary sequence constraint based on multi-species genome alignments and genome-wide maps of epigenomic marks and transcription factor binding provide important complementary information for understanding the human genome and genetic variation. Here we developed the Constrained Non-Exonic Predictor (CNEP) to quantify the evidence of each base in the genome being in an evolutionarily constrained non-exonic element from an input of over 60,000 epigenomic and transcription factor binding features. We find that the CNEP score outperforms baseline and related existing scores at predicting evolutionarily constrained non-exonic bases from such data. However, a subset of them are still not well predicted by CNEP. We developed a complementary Conservation Signature Score by CNEP (CSS-CNEP) that is predictive of those bases. We further characterize the nature of constrained non-exonic bases with low CNEP scores using additional types of information. CNEP and CSS-CNEP are resources for analyzing constrained non-exonic bases in the genome.

ZNF423 orthologs are highly constrained in vertebrates but show domain-level plasticity across invertebrate lineages

ZNF423 encodes 30 C2H2 zinc fingers that bind DNA and a variety of lineage- and signal-dependent transcription factors. ZNF423 genetic variants are proposed to cause neurodevelopmental and ciliopathy-related disorders in humans. Mouse models show midline brain defects, including cerebellar vermis hypoplasia, and defects in adipogenesis. Here I show strong protein sequence constraint among 165 vertebrate orthologs. In contrast, orthologs from invertebrate lineages, spanning larger time intervals, show substantial differences in zinc finger number, arrangement, and identity. A terminal zinc finger cluster common among other lineages was independently lost in vertebrates and insects. Surprisingly, a moderately-constrained non-C2H2 sequence with potential to form a C4-class zinc finger is a previously-unrecognized conserved feature of nearly all identified homologs. These results highlight evolutionary dynamics of a likely signal integration node across species with distinct developmental strategies and body plans. Functions of the newly identified C4-like sequence and lineage-specific fingers remain to be studied.