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It is well known that the complete multipartite graphs can not be determined by their adjacency spectra. But in this paper, we prove that they can be determined by their distance spectra, which confirms the conjecture proposed by Lin, Hong,…
We consider finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies. They can be applied to the study of modular forms and to isogeny based cryptography. The main result of this…
In this paper we analyze the spectral gap of a weighted graph which is the difference between the smallest positive and largest negative eigenvalue of its adjacency matrix. Such a graph can represent e.g. a chemical organic molecule. Our…
The graph alignment problem, which considers the optimal node correspondence across networks, has recently gained significant attention due to its wide applications. There are graph alignment methods suited for various network types, but we…
Graph energy is the energy of the matrix representation of the graph, where the energy of a matrix is the sum of singular values of the matrix. Depending on the definition of a matrix, one can contemplate graph energy, Randi\'c energy,…
The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged. If e={i,j} is an edge of G then G*e=((G*i)*j)*i is…
We study the problem of connecting the parts of a multipartite graph using a minimum number of edges under a matching constraint. We introduce interconnection trees, defined as matchings whose projections onto the quotient graph form a…
Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal…
This paper analyzes the problem of assigning weights to edges incrementally in a dynamic complete bipartite graph consisting of producer and consumer nodes. The objective is to minimize the overall cost while satisfying certain constraints.…
The maximum-weight matching problem and the behavior of its energy landscape is numerically investigated. We apply a perturbation method adapted from the analysis of spin glasses. This gives inside into the complexity of the energy…
In 2009, Kong, Wang, and Lee introduced the problem of finding the edge-balanced index sets ($EBI$) of complete bipartite graphs $K_{m,n}$, where they examined the cases $n=1$, $2$, $3$, $4$, $5$ and the case $m=n$. Since then the problem…
We applied tensor network contraction algorithms to compute the hard-core lattice gas model, i.e., the enumeration of independent sets on grid graphs. We observed the influence of surface effect and parity effect on the enumeration (and…
Graph Sampling provides an efficient yet inexpensive solution for analyzing large graphs. While extracting small representative subgraphs from large graphs, the challenge is to capture the properties of the original graph. Several sampling…
The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts.…
Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…
Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered…
We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by this relations. We first give a Borel reduction from embeddability on graphs to…
We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha}{1-\alpha}$ and the…
Iterated Graph Systems (IGS) transplant ideas from fractal geometry into graph theory. Building on this framework, we extend Edge IGS from the primitive to the reducible setting. Within this broader context, we formulate rigorous…
We study the vertex classification problem on a graph whose vertices are in $k\ (k\geq 2)$ different communities, edges are only allowed between distinct communities, and the number of vertices in different communities are not necessarily…