Related papers: Subspace Sum Graph of a Vector Space

We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of…

For a group $G$ and a subset $X$ of $G$, the commuting graph of $X$, denoted by $\Gamma(G,X)$ is the graph whose vertex set is $X$ and any two vertices $u$ and $v$ in $X$ are adjacent if and only if they commute in $G$. In this article,…

For a graph $G=(V,E)$, we call a subset $ S\subseteq V \cup E$ a total mixed dominating set of $G$ if each element of $V \cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $\gamma_{tm}(G)$ of…

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space…

We prove that if $G$ is a graph and $r_1, ..., r_k \in \mathbb{Z}_{\geq 0}$ such that $\sum_{i=1}^k r_i \geq \Delta(G) + 2 - k$ then $V(G)$ can be partitioned into sets $V_1, ..., V_k$ such that $\Delta(G[V_i]) \leq r_i$ and $G[V_i]$…

Let $\G(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\sum d_i^2$ denote the sum of the squares of the degrees, $d_1, >..., d_v$, of the vertices of $G$. It is known that the maximum value of $P_2(G)$…

The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…

For a graph $G=(V,E)$, and a symplectic vector space $(W, \left<\cdot,\cdot\right>)$, we define a variety $X(G,W)$ consisting of all functions $w:V\to W$ satisfying $\left<w(u), w(v)\right> = 0$ for any edge $\{u,v\}$ in $G$. We study the…

We study to which extent the family of pairs of subspaces of a vector space related to each other via intersection properties determines the vector space. In another language, we study to which extent the family of vertices of the building…

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…

The main aim of this paper is to generalize the concept of vector space by the hyperstructure. We generalize some definitions such as hypersubspaces, linear combination, Hamel basis, linearly dependence and linearly independence. A few…

For a simple graph G = (V, E), a coloring of vertices of G using two colors, say red and blue, is called a quasi neighborhood balanced coloring if, for every vertex of the graph, the number of red neighbors and the number of blue neighbors…

A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is…

The symplectic graph Sp(2d, q) is the collinearity graph of the symplectic space of dimension 2d over a finite field of order q. A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1, lambda_2 ,m,n) if…

The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent…

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and…

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only…

Suppose G is an n-vertex simple graph with vertex set {v1,..., vn} and d(i), i = 1,..., n, is the degree of vertex vi in G. The ISI matrix S(G) = [sij] of G is a square matrix of order n and is defined by sij = d(i)d(j)/d(i)+d(j) if the…

Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R^d such that two vertices are mapped to orthogonal vectors if and only if they are not neighbors. The minimum vector rank of a graph is the…

This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss…