Related papers: Graphs with few trivial characteristic ideals
In this paper we develop three characterizations for isomorphism of graphs. The first characterization is obtained by associating certain bitableaux with the graphs. We order these bitableaux by suitably defined lexicographic order and…
A 2-factor of a graph is a 2-regular spanning subgraph. For a graph $G$ and an independent set $I$ of $G$, let $\delta_G(I)$ denote the minimum degree of vertices contained in $I$. We show that (1) if every independent set $I$ of $G$…
Directed graphs have long been used to gain understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Fruct's…
The family of cycle completable graphs has several cryptomorphic descriptions, the equivalence of which has heretofore been proven by a laborious implication-cycle that detours through a motivating matrix completion problem. We give a…
Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a…
Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined…
Given a graph $G=(V, E)$, its generalized Laplacian matrix is given by \[ L(G,X_G)_{u,v}= \begin{cases} x_u&\text{if }u=v,\\ -m_{uv}&\text{if }u\neq v, \end{cases} \] where $X_G=\{x_u\, | \, u\in V(G)\}$ is a set of indeterminates and…
The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut…
We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph…
We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set…
Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by…
For a given graph consider a pair of disjoint matchings the union of which contains as many edges as possible. Furthermore, consider the relation of the cardinalities of a maximum matching and the largest matching in those pairs. It is…
A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k< n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph is minimal if for every edge, the deletion of…
Let $G$ be a finite group, $n$ a positive integer. $\pi(n)$ denotes the set of all prime divisors of $n$ and $\pi(G)=\pi(|G|)$. The prime graph $\Gamma(G)$ of $G$, defined by Grenberg and Kegel, is a graph whose vertex set is $\pi(G)$, two…
Graphs derived from groups are a widely studied class of graphs, motivated by their highly symmetric structure. In particular, G-graphs offer an easy and interesting alternative construction of semi-symmetric graphs. After recalling the…
We classify the ultrahomogeneous complete 3-edge-coloured graphs (3-graphs) with simple theory. This extends Lachlan's result (a corollary of the Effective Classification Theorem for stable structures) classifying the stable homogeneous…
We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial.
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…
The critical group of a finite graph is an abelian group defined by the Smith normal form of the Laplacian. We determine the the critical groups of the Peisert graphs, a certain family of strongly regular graphs similar to, but different…
Chirality is one of the important assymmetrical property in wide area of natural science, which has been studied to predict molecular behavior. One of good methods to analyze molecules with complex structures is representing them as graphs…