Related papers: Integral trees with given nullity
A graph is $\alpha$-excellent if every vertex of the graph is contained in some maximum independent set of the graph. In this paper, we present two characterizations of the $\alpha$-excellent $2$-trees.
The proper thinness of a graph is an invariant that generalizes the concept of a proper interval graph. Every graph has a numerical value of proper thinness and the graphs with proper thinness~1 are exactly the proper interval graphs. A…
A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\geq 2$). Let $S$ be a vertex subset of $G$ such that $\alpha_{G}(S)…
A graph in which all minimal zero forcing sets are in fact minimum size is called ``well-forced." This paper characterizes well-forced trees and presents an algorithm for determining which trees are well-forced. Additionally, we…
The status of a vertex $x$ in a graph is the sum of the distances between $x$ and all other vertices. Let $G$ be a connected graph. The status sequence of $G$ is the list of the statuses of all vertices arranged in nondecreasing order. $G$…
We prove that for any positive integer $k$, the edges of any graph whose fractional arboricity is at most $k + 1/(3k+2)$ can be decomposed into $k$ forests and a matching.
Mining for trees in a graph is shown to be NP-complete.
The induced arboricity of a graph $G$ is the smallest number of induced forests covering the edges of $G$. This is a well-defined parameter bounded from above by the number of edges of $G$ when each forest in a cover consists of exactly one…
A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S|…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…
Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs…
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…
Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,…
The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…
A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial…
An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. It is well known that a graph $G$ has exactly two main eigenvalues if and only if there exists a unique pair of…
The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a…
One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph…
A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet…