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A spanning tree of a graph without no vertices of degree $2$ is called a {\it homeomorphically irreducible spanning tree} (or a {\it HIST}) of the graph. Albertson, Berman, Hutchinson and Thomassen~[J. Graph Theory {\bf 14} (1990), 247--258] gave a minimum degree condition for the existence of a HIST, and recently, Ito and Tsuchiya~[J. Graph Theory {\bf 99} (2022), 162--170] found a sharp degree-sum condition for the existence of a HIST. In this paper, we refine these results, and extend the first one to a spanning tree in which no vertex other than the endvertices has small degree., Comment: 18 pages, 5 figures

Reducing antibiotic usage in livestock animals has become an urgent issue worldwide to prevent antimicrobial resistance. Here, abuse of chlortetracycline (CTC), a versatile antibacterial agent, on the performance, blood components, fecal microbiota, and organic acid concentration in calves was investigated. Japanese Black calves were fed milk replacer containing CTC at 10 g/kg (CON) or 0 g/kg (EXP). Growth performance was not affected by CTC administration. However, CTC administration altered the correlation between fecal organic acids and bacterial genera. Machine learning methods such as association analysis, linear discriminant analysis, and energy landscape analysis revealed that CTC administration affected according to certain rules the population of various types of fecal bacteria. It is particularly interesting that the population of several methane-producing bacteria was high in the CON, and that of Lachnospiraceae, a butyrate-producing bacteria, was high in the EXP at 60 d of age. Furthermore, statistical causal inference based on machine learning data estimated that CTC treatment affects the entire intestinal environment, inhibiting butyrate production for growth and biological defense, which may be attributed to methanogens in feces. Thus, these observations highlight the multiple harmful impacts of antibiotics on intestinal health and the potential production of greenhouse gas in the calves.

There is a long line of research in the literature dedicated to word-representable graphs, which generalize several important classes of graphs. However, not much is known about word-representability of split graphs, another important class of graphs. In this paper, we show that threshold graphs, a subclass of split graphs, are word-representable. Further, we prove a number of general theorems on word-representable split graphs, and use them to characterize computationally such graphs with cliques of size 5 in terms of 9 forbidden subgraphs, thus extending the known characterization for word-representable split graphs with cliques of size 4. Moreover, we use split graphs, and also provide an alternative solution, to show that gluing two word-representable graphs in any clique of size at least 2 may, or may not, result in a word-representable graph. The two surprisingly simple solutions provided by us answer a question that was open for about ten years.

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0\rightarrow v_1\rightarrow \cdots\rightarrow v_k$ either there is no edge between $v_0$ and $v_k$, or $v_i\rightarrow v_j$ is an edge for all $0\leq i

We propose a method for generalizing the Ising model in magnetic fields and calculating the partition function (exact solution) for the Ising model of an arbitrary shape. Specifically, the partition function is calculated using matrices that are created automatically based on the structure of the system. By generalizing this method, it becomes possible to calculate the partition function of various crystal systems (network shapes) in magnetic fields when N (scale) is infinite. Furthermore, we also connect this method for finding the solution to the Ising model in magnetic fields to a method for finding the solution to Bayesian networks in information statistical mechanics (applied to data mining, machine learning, and combinatorial optimization).

A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every $\frac{3}{2}$-tough $2K_2$-free graph with at least three vertices has a spanning trail with maximum degree at most $4$. In this paper, we confirm this conjecture. We also provide examples for all $t < \frac{5}{4}$ of $t$-tough graphs that do not have a spanning trail with maximum degree at most $4$.

This study presents a generalization for a method examining the correlation function of an arbitrary system with interactions in an Ising model to obtain a value of correlation between two arbitrary points on a network. The establishment of a network clarifies the type of calculations necessary for the correlation values between secondary and tertiary nodes. Moreover, it is possible to calculate the correlation values of the target that are interlinked in a complex manner by proposing a network analysis method to express the same as a network with mutual linkages between the target of each field.

In this paper, we consider coloring of graphs under the assumption that some vertices are already colored. Let $G$ be an $r$-colorable graph and let $P\subset V(G)$. Albertson [J.\ Combin.\ Theory Ser. B \textbf{73} (1998), 189--194] has proved that if every pair of vertices in $P$ have distance at least four, then every $(r+1)$-coloring of $G[P]$ can be extended to an $(r+1)$-coloring of $G$, where $G[P]$ is the subgraph of $G$ induced by $P$. In this paper, we allow $P$ to have pairs of vertices of distance at most three, and investigate how the number of such pairs affects the number of colors we need to extend the coloring of $G[P]$. We also study the effect of pairs of vertices of distance at most two, and extend the result by Albertson and Moore [J.\ Combin.\ Theory Ser. B \textbf{77} (1999) 83--95].