Related papers: Regular Steinhaus graphs of odd degree

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…

This paper addresses the challenge of spectral analysis and structural investigation for graphs that are not distance-regular, where computing the spectrum using standard methods based on equitable and orbit partitions can be complex. Our…

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that…

A square (0,1)-matrix X of order n > 0 is called fully indecomposable if there exists no integer k with 0 < k < n, such that X has a k by n-k zero submatrix. A stable set of a graph G is a subset of pairwise nonadjacent vertices. The…

In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in…

The joint degree matrix of a graph gives the number of edges between vertices of degree i and degree j for every pair (i,j). One can perform restricted swap operations to transform a graph into another with the same joint degree matrix. We…

A mixed graph can be seen as a type of digraph containing some edges (two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures…

The paper gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from $x$ to $y$ is equal to the complex unity…

For $k,s\geq2$, the $s$-stable Kneser graphs are the graphs with vertex set the $k$-subsets $S$ of $\{1,\ldots,n\}$ such that the circular distance between any two elements in $S$ is at least $s$ and two vertices are adjacent if and only if…

Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…

A folklore result attributed to P\'olya states that there are $(1 + o(1))2^{\binom{n}{2}}/n!$ non-isomorphic graphs on $n$ vertices. Given two graphs $G$ and $H$, we say that $G$ is a unique subgraph of $H$ if $H$ contains exactly one…

A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $\Sigma$ (that is,…

A nut graph is a simple graph whose adjacency matrix has the eigenvalue~0 with multiplicity~1 such that its corresponding eigenvector has no zero entries. Motivated by a question of Fowler et al.~[\emph{Disc. Math. Graph Theory} 40 (2020),…

Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The sum of all Steiner distances on sets of size $k$ is called the…

Let $n \geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n…

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented…

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…

We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let $\Gamma$ be a graph with vertex set $V$, diameter $D$, adjacency matrix $A$, and…

In 2002, D. Fon-Der-Flaass constructed a prolific family of strongly regular graphs. In this paper, we prove that for infinitely many natural numbers $n$, this family contains $n^{\Omega(n^{2/3})}$ strongly regular $n$-vertex graphs $X$…