Related papers: Independent sets in polarity graphs

Given a partition $V_1 \sqcup V_2 \sqcup \dots \sqcup V_m$ of the vertex set of a graph, we are interested in finding multiple disjoint independent sets that contain the correct fraction of vertices of each $V_j$. We give conditions for the…

The independence number of a graph G, denoted by alpha(G), is the cardinality of an independent set of maximum size in G, while mu(G) is the size of a maximum matching in G, i.e., its matching number. G is a Konig-Egervary graph if its…

The article considers an inhomogeneous Erd\H{o}s-R\"enyi random graph on $\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made independent…

For a fixed positive integer $t$, we consider the graph colouring problem in which edges at distance at most $t$ are given distinct colours. We obtain sharp lower bounds for the distance-$t$ chromatic index, the least number of colours…

Let $X$ be $k$-regular graph on $v$ vertices and let $\tau$ denote the least eigenvalue of its adjacency matrix $A(X)$. If $\alpha(X)$ denotes the maximum size of an independent set in $X$, we have the following well known bound: \[…

The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and…

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are…

Let $\sigma$ be a partition of the positive integer $r$. A $\sigma$-hypergraph $H=H(n,r,q|\sigma)$ is an $r$-uniform hypergraph on $nq$ vertices which are partitioned into $n$ classes $V_1, V_2, \ldots, V_n$ each containing $q$ vertices. An…

A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao. They show that for large graphs there exist real orthonormal basis of eigenfunctions of the adjacency matrix such…

We examine the Maximum Independent Set Problem in an undirected graph. The main result is that this problem can be considered as the solving the same problem in a subclass of the weighted normal twin-orthogonal graphs. The problem is…

In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…

Let G be a simple graph with vertex set V(G). A set S is independent if no two vertices from S are adjacent. The graph G is known to be a Konig-Egervary if alpha(G)+mu(G)= |V(G)|, where alpha(G) denotes the size of a maximum independent set…

Given a graph G of order n and size m, let s(G)= sum|d(u)-2m/n|, where the sum is taken over all vertices u of G. We investigate upper and lower bounds on eigenvalues of G in terms of s(G).

Given nonnegative integers, $s$ and $k$, an $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of $V_G$ such that $G[A]$ and $\overline{G[B]}$ are complete multipartite graphs with at most $s$ and $k$ parts, respectively. If $s$…

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether…

An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $\alpha{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for…

Given a graph $G$, we form a random subgraph $G_p$ by including each edge of $G$ independently with probability $p$. We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A set $I_0(G) \subseteq V(G)$ is a vertex independent set if no two vertices in $I_0(G)$ are adjacent in $G$. We study $\alpha_1(G)$, which is the maximum cardinality of a set…

A polyhedral graph is a $3$-connected planar graph. We find the least possible order $p(k,a)$ of a polyhedral graph containing a $k$-independent set of size $a$ for all positive integers $k$ and $a$. In the case $k = 1$ and $a$ even, we…

This paper consists of a few results, discovered and proved during the 2012-2013 research group at Eastern Oregon University. Inertia tables are a visual representation of the possible inertias of a given graph. The inertia of a graph…