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We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs…

We study notions of hyperuniformity for invariant locally square-integrable point processes in regular trees. We show that such point processes are never geometrically hyperuniform, and if the diffraction measure has support in the…

For $0\leq \alpha < 1$, the $\mathcal{A}_{\alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_{\alpha}(G):=\alpha \mathcal{D}(G)+(1-\alpha) \mathcal{A}(G)$, where…

Unigraphs are graphs identifiable up to isomorphism from their degree sequences. Given a class $\mathcal{A}$ of graphs, we define the class of $\mathcal{A}$-unigraphs to be graphs identifiable from degree sequence and membership in…

We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the…

We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r >= 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from…

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let $G$ be a graph and $G^m$ be the $m$-th power (hypergraph) of $G$. The spectrum of $G$ is referring to its adjacency…

Let $G(n,k)$ be the class of clique trees on $n$ vertices and zero forcing number $k$, where $\left \lfloor \frac{n}{2} \right \rfloor + 1 \le k \le n-1$ and each block is a clique of size at least $3$. In this article, we proved the…

A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is…

The spectral properties of traditional (dyadic) graphs, where an edge connects exactly two vertices, are widely studied in different applications. These spectral properties are closely connected to the structural properties of dyadic…

The Harary index of a graph $G$ is recently introduced topological index, defined on the reverse distance matrix as $H(G)=\sum_{u,v \in V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the length of the shortest path between two distinct vertices…

The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as $v_1, v_2, \ldots, v_n$, so that for each $2 \le i \le n$, vertex $v_i$ is either adjacent…

A sequence of vertices $(v_1,\, \dots , \,v_k)$ of a graph $G$ is called a {\it dominating closed neighborhood sequence} if $\{v_1,\, \dots , \,v_k\}$ is a dominating set of $G$ and $N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]$ for every $i$.…

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

We give some graph theoretical formulas for the trace $Tr_k(\mathbb {T})$ of a tensor $\mathbb {T}$ which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017)…

Chv\'atal showed that for any tree $T$ with $k$ edges the Ramsey number $R(T,n)=k(n-1)+1$ ("Tree-complete graph Ramsey numbers." Journal of Graph Theory 1.1 (1977): 93-93). For $r=3$ or $4$, we show that, if $T$ is an $r$-uniform…

The $p$-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a…

Let $\mathcal{H}^{(r)}_n$ be the set of all connected $r$-graphs with given size $n$. In this paper, we investigate the effect on the spectral radius of $r$-uniform hypergraphs by grafting or contracting an edge and then give the ordering…

We define $k$-dimensional digraphs and initiate a study of their spectral theory. The $k$-dimensional digraphs can be viewed as generating graphs for small categories called $k$-graphs. Guided by geometric insight, we obtain several new…