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A $k$-tree is a spanning tree in which every vertex has degree at most $k$. In this paper, we provide a sufficient condition for the existence of a $k$-tree in a connected graph with fixed order in terms of the adjacency spectral radius and…
We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM…
Let $G$ be a graph and $T$ be a spanning tree of $G$. We use $Q(G)=D(G)+A(G)$ to denote the signless Laplacian matrix of $G$, where $D(G)$ is the diagonal degree matrix of $G$ and $A(G)$ is the adjacency matrix of $G$. The signless…
We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures…
In the past decades, graphs that are determined by their spectrum have received more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An…
A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We…
We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum…
We consider the Laplacian on a rooted metric tree graph with branching number $ K \geq 2 $ and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely…
Consider two quantum graphs with the standard Laplace operator and non-Robin type boundary conditions at all vertices. We show that if their eigenvalue-spectra agree everywhere aside from a sufficiently sparse set, then the…
Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf…
We investigate spectral properties of quantum graphs in the form of a periodic chain of rings with a connecting link between each adjacent pair, assuming that wave functions at the vertices are matched through conditions manifestly…
Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on…
We show how the spectrum of a graph Laplacian changes with respect to a certain type of rank-one perturbation. We apply our finding to give new short proofs of the spectral version of Kirchhoff's Matrix Tree Theorem and known derivations…
We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…
The spectral gap for Laplace operators on metric graphs is investigated in relation to graph's connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs…
We introduce a new notion of graph sparsificaiton based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to…
We introduce and study a generalization of conformal rigidity for graphs. A graph is $k$-conformally rigid if the uniform edge weights simultaneously maximize the sum of the $k$ smallest nontrivial Laplacian eigenvalues and minimize the sum…
Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity…
A graph is called $K$-almost regular if its maximum degree is at most $K$ times the minimum degree. Erd\H{o}s and Simonovits showed that for a constant $0< \varepsilon< 1$ and a sufficiently large integer $n$, any $n$-vertex graph with more…
Cliquewidth is a dense analogue of treewidth. It can be deduced from recent results by Hickingbotham [arXiv:2501.10840] and Nguyen, Scott, and Seymour [arXiv:2501.09839] that graphs of bounded cliquewidth are quasi-isometric to graphs of…