Related papers: Graphs with few trivial characteristic ideals
We study how few pairwise distinct longest cycles a regular graph can have under additional constraints. For each integer $r \geq 5$, we give exponential improvements for the best asymptotic upper bounds for this invariant under the…
Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of $G$ the set of colors appearing on the edges incident with $v$. The palette index of $G$ is defined as the minimum number of such distinct sets, taken over all…
While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult…
A graph $\Gamma$ is basic if Aut$\Gamma$ has no normal subgroup $N\ne1$ such that $\Gamma$ is a normal cover of the normal quotient graph $\Gamma_N$. In this paper, we completely determine the basic normal quotient graphs of all connected…
A strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that every infinite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson.
The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants…
Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum.…
A non-complete graph $G$ is said to be $t$-tough if for every vertex cut $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. The toughness $\tau(G)$ of the graph $G$ is the maximum value of $t$ such that $G$…
We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological…
Let $\alpha=\alpha(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is…
A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors of…
In this study we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This give an alternative characterization of triangulated graphs. Our method is based on…
We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…
In this paper, we aim to address the open questions raised in various recent papers regarding characterization of circulant graphs with three or four distinct eigenvalues in their spectra. Our focus is on providing characterizations and…
Directed acyclic graphs whose nodes are all the divisors of a positive integer $n$ and arcs $(a,b)$ defined by $a$ divides $b$ are considered. Fourteen graph invariants such as order, size, and the number of paths are investigated for two…
For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there…
By the Grunbaum-Aksenov Theorem (extending Grotzsch's Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such…
We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several…
The critical group of a connected graph is closely related to the graph Laplacian, and is of high research value in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. In this paper, we study the…
The classical Whitney's 2-Isomorphism Theorem describes the families of graphs having the same cycle matroid. In this paper we describe the families of graphs having the same truncated cycle matroid and prove, in particular, that every…