Related papers: $k$-NIM trees: Characterization and Enumeration

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result…

This note derives asymptotic upper and lower bounds for the number of planted plane trees on $n$ nodes assigned labels from the set $\{1,2,\ldots, k\}$ with the restriction that on any path from the root to a leaf, the labels must strictly…

We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs In particular, we consider the case of thick forests, which we…

Given a graph $G=(V,E)$ with non-negative real edge lengths and an integer parameter $k$, the Min-Max k-Tree Cover problem seeks to find a set of at most $k$ subtrees of $G$, such that the union of the trees is the vertex set $V$. The…

The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs in open. The complexities of recognising many subclasses of SOGs are known.…

We prove a Sidorenko-type inequality for directed trees: for every oriented tree $T$ on $k$ vertices and every finite directed graph $G$, the homomorphism count hom$(T,G)$ is bounded above by the maximum of the two pure star counts…

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are…

Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected…

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ such that any nontrivial path whose endpoints have the same label contains a larger label. The least $k$ for which $G$ has a $k$-ranking is the ranking number…

Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $\rho(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{0,d_T(v)-k\}$. The total $k$-excess $\mbox{te}(T,k)$ is…

We examine several types of visibility graphs in which sightlines can pass through $k$ objects. For $k \geq 1$ we bound the maximum thickness of semi-bar $k$-visibility graphs between $\lceil \frac{2}{3} (k + 1) \rceil$ and $2k$. In…

We study higher-dimensional analogues of graph-theoretic trees within the class of pure n-simplicial complexes. Focusing on the case m = n-1 in Dewdney's (m, n)-tree framework, we introduce refined notions of path and circuit sequences that…

Estimating phylogenetic trees, which depict the relationships between different species, from aligned sequence data (such as DNA, RNA, or proteins) is one of the main aims of evolutionary biology. However, tree reconstruction criteria like…

The underlying graph $G$ of a symmetric matrix $M=(m_{ij})\in \mathbb{R}^{n\times n}$ is the graph with vertex set $\{v_1,\ldots,v_n\}$ such that a pair $\{v_i,v_j\}$ with $i\neq j$ is an edge if and only if $m_{ij}\neq 0$. Given a graph…

The number of spanning trees in a graph $G$ is the total number of distinct spanning subgraphs of $G$ that are trees. In this paper we characterize the unique graph with a prescribed vertex (resp. edge) connectivity, minimum degree and…

Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n,k) is the graph with vertex set $\binom{[n]}{k}$, such that two vertices are adjacent if they are disjoint. We…

The index of a graph is the largest eigenvalue of its adjacency matrix. A starlike is a tree having a unique vertex of degree $r>2$. We show how to order the starlike trees with $n>3$ by their indices. In particular, the indices of starlike…

For natural numbers $k<n$ we study the graphs $T_{n,k}:=K_{k}\lor\overline{K_{n-k}}$. For $k=1$, $T_{n,1}$ is the star $S_{n-1}$. For $k>1$ we refer to $T_{n,k}$ as a \emph{graph of pyramids}. We prove that the graphs of pyramids are…

A graph $G=(V,E)$ is a $k$-leaf power if there is a tree $T$ whose leaves are the vertices of $G$ with the property that a pair of leaves $u$ and $v$ induce an edge in $G$ if and only if they are distance at most $k$ apart in $T$. For $k\le…

A graph $G$ is a $k$-leaf power if there exists a tree $T$ whose leaf set is $V(G)$, and such that $uv \in E(G)$ if and only if the distance between $u$ and $v$ in $T$ is at most $k$. The graph classes of $k$-leaf powers have several…