Related papers: Polynomial and spectra factorization of graphs obt…
If $G$ is a strongly connected finite directed graph, the set $\mathcal{T}G$ of rooted directed spanning trees of $G$ is naturally equipped with a structure of directed graph: there is a directed edge from any spanning tree to any other…
In this article we continue our analysis of Schr\"odinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule…
The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed by Wang, Wang, and Zhu (European J. Combin., 2023), which…
In this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and…
We develop an operator algebraic framework for generalized Cardano polynomials and show how their structure naturally leads to an operator formulation of Cardano method that is compatible with tools and concepts from quantum information…
In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic…
A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees,…
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
Graph-based semantic representations are valuable in natural language processing, where it is often simple and effective to represent linguistic concepts as nodes, and relations as edges between them. Several attempts has been made to find…
An algebraic formalism, developped with V. Glaser and R. Stora for the study of the generalized retarded functions of quantum field theory, is used to prove a factorization theorem which provides a complete description of the generalized…
For a simple graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), \alpha\in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees of…
The bipartition polynomial of a graph is a generalization of many other graph polynomials, including the domination, Ising, matching, independence, cut, and Euler polynomial. We show in this paper that it is also a powerful tool for proving…
Observational data usually comes with a multimodal nature, which means that it can be naturally represented by a multi-layer graph whose layers share the same set of vertices (users) with different edges (pairwise relationships). In this…
Chromatic polynomials have been studied extensively, giving us results such as the Fundamental Reduction Theorem and closed formulas for the chromatic polynomials of common classes of graphs. Though, none of those extend to the context of…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
The main object of this work is the top-dimensional Laplacian operator of a simplicial complex $K$. We study its spectral limiting behavior under a given non-trivial subdivision procedure $\text{div}$. It will be shown that in case…
Let $G, H_{i}$ be simple graphs with $n=|V(G)|$, $m=|E(G)|$ and $i=1, 2, \ldots, n(m)$. The generalized corona, denoted $G\tilde{o}\wedge^{n}_{i=1} H_{i}$, is the graph obtained by taking one copy of graphs $G, H_{1},\ldots, H_{n}$ and…
We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star…
Expectation propagation is a general approach to fast approximate inference for graphical models. The existing literature treats models separately when it comes to deriving and coding expectation propagation inference algorithms. This comes…