Related papers: Fat Hoffman graphs with smallest eigenvalue greate…

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result…

In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The…

The Hamming graph $H(n,q)$ is the graph whose vertices are the words of length $n$ over the alphabet $\{0,1,\ldots,q-1\}$, where two vertices are adjacent if they differ in exactly one coordinate. The adjacency matrix of $H(n,q)$ has $n+1$…

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two…

In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $\lambda\leq -1$, we show that the…

For two graphs $G$ and $F$, we say that $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ as a subgraph and one could join all the nonadjacent pairs of vertices of $G$ in some order so that a new copy of $F$ is created at each…

We provide a combinatorial characterization of all testable properties of $k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$ is testable if there is a randomized algorithm which makes a bounded number of edge…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

We consider the graph whose vertex set is a conjugacy class ${\mathcal C}$ consisting of finite-rank self-adjoint operators on a complex Hilbert space $H$. The dimension of $H$ is assumed to be not less than $3$. In the case when operators…

Let $S_n$ denote the symmetric group on $n$ letters. The $k$-point fixing graph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $S_n$ and two vertices $g,h$ of $\mathcal{F}(n,k)$ are joined by an edge, if and only if $gh^{-1}$…

A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $\mathrm{sat}(n, H, F)$ which denotes the minimum number of copies of $H$ that an $F$-saturated graph on $n$…

In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply…

An $n$-vertex graph $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ and there exists an ordering of all edges in $E(K_n) \setminus E(G)$ such that, when added one at a time, each edge creates a new copy of $F$. The minimum size…

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is…

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular,…

We investigate Hadwiger's conjecture for graphs with no stable set of size 3. Such a graph on at least 2t-1 vertices is not t-1 colorable, so is conjectured to have a $K_t$ minor. There is a strengthening of Hadwiger's conjecture in this…

A graph is hypohamiltonian if it is non-Hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay,…

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…

In a recent paper, Gerbner, Patk\'{o}s, Tuza and Vizer studied regular $F$-saturated graphs. One of the essential questions is given $F$, for which $n$ does a regular $n$-vertex $F$-saturated graph exist. They proved that for all…

This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the $k$-th largest…