Inclusion Graph Research Articles

$W^*$-representations of subfactors and restrictions on the Jones index

A W^* - representation of a \mathrm{II}_1 subfactor N\subset M with finite Jones index, [M:N]<\infty , is a non-degenerate commuting square embedding of N\subset M into an inclusion of atomic von Neumann algebras \oplus_{i\in I} \mathcal{B}(\mathcal K_i)=\mathcal N \subset^{\mathcal E} \mathcal M=\oplus_{j\in J} \mathcal{B}(\mathcal H_j) . We undertake here a systematic study of this notion, first introduced by the author in 1992, giving examples and considering invariants such as the (bipartite) inclusion graph \Lambda_{\mathcal N \subset \mathcal M} , the coupling vector (\dim({}_M\mathcal H_j))_j and the RC-algebra (relative commutant) M'\cap \mathcal N , for which we establish some basic properties. We then prove that if N\subset M admits a W^* -representation \mathcal N\subset^{\mathcal E}\mathcal M , with the expectation \mathcal E preserving a semifinite trace on \mathcal M , such that there exists a norm one projection of \mathcal M onto M commuting with \mathcal E , a property of N\subset M that we call weak injectivity/amenability , then [M:N] equals the square norm of the inclusion graph \Lambda_{\mathcal N \subset \mathcal M} .

Automorphisms of symplectic totally isotropic subspace inclusion graph

Recently, symplectic totally isotropic subspace inclusion graph [Formula: see text] on a [Formula: see text]-dimensional symplectic space [Formula: see text] was introduced in [J. He, S. Zhang and G. Zhang, Symplectic totally isotropic subspace inclusion graph, Linear Multilinear Algebra (2022), doi:10.1080/03081087.2022.2112016], which is a graph whose vertices are all totally isotropic subspaces of [Formula: see text] and two distinct vertices are adjacent if and only if one is contained in the other. In that paper, the authors studied the diameter, girth, clique number and chromatic number of [Formula: see text] and also studied some other properties of [Formula: see text]. In this paper, we determine the automorphism group of [Formula: see text].

On some properties of vector space based graphs

In this paper, we study some problems related to subspace inclusion graph In⁡(V) and subspace sum graph G(V) of a finite-dimensional vector space V. Namely, we prove that In⁡(V) is a Cayley graph as well as Hamiltonian when the dimension of V is 3. We also find the exact value of independence number of G(V) when the dimension of V is odd. The above two problems were left open in previous works in the literature. Moreover, we prove that the determining numbers of In⁡(V) and G(V) are bounded above by 6. Finally, we study some forbidden subgraphs of these two graphs.

Symplectic totally isotropic subspace inclusion graph

Symplectic totally isotropic subspace inclusion graph on a finite-dimensional symplectic space is a graph whose vertices are all totally isotropic subspaces of and two distinct vertices are adjacent if and only if one is contained in other. In this paper, the diameter, girth, clique number, chromatic number of are studied. Furthermore, a necessary and sufficient condition is provided for to be Eulerian. It is shown that if two symplectic totally isotropic subspace inclusion graphs on symplectic space are isomorphic, then the two subspace inclusion graphs on corresponding vector space are also isomorphic. It is also shown that is non-planar.

Study of some properties of complement of open subset inclusion graph of a topological space

In the recent paper, authors introduced a graph topological structure, called as open subset inclusion graph 𝚥(𝜏) of a topological space (𝑋, 𝜏) on a finite set 𝑋 and discussed some important properties of this graph. In this paper, we discuss some properties of the graph 𝚥(𝜏) 𝑐 . It is shown that, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| ≤ 3, then the graph 𝚥(𝜏) 𝑐 is bipartite, and if |𝑋| = 2, then the graph 𝚥(𝜏) 𝑐 is regular & complete bipartite. Moreover, if 𝜏 is a discrete topology defined on nonempty set 𝑋 with |𝑋| = 2 or |𝑋| = 3 then it is shown that the graph 𝚥(𝜏) 𝑐 is Hamiltonian, vertex-transitive, edge-transitive and has a perfect matching. We also provide exact value of the independence number, vertex connectivity and edge connectivity of the graph 𝚥(𝜏) 𝑐 of a discrete topology defined on nonempty set𝑋 with |𝑋| = 2 or |𝑋| = 3. Main finding of this work is that, if (𝑋, 𝜏) is a discrete topological space with |𝑋| = 2 or |𝑋| = 3 then it is shown that 𝚥(𝜏) 𝑐 is distance-transitive graph and distance regular graph.

Some results of inclusion graph of a topology

In this paper, we introduce a graph topological structure called Inclusion Graph of a Topology . The diameter, girth, connectivity, maximal independent sets, different variants of domination number and chromatic number , edge and vertex connectivity of are studied. Moreover, we derive the upper and lower bonds of clique number and a chromatic number of the graph .

Automorphism group of the open subset inclusion graph of a topological space

In this paper, we determine automorphisms of j(τ) and prove some results on j(τ),when topological space (X, τ) is finite. Also we determine necessary and sufficient condition for the open subset inclusion graphs of (X, τ 1) and (X, τ 2) to become isomorphic. Moreover, the topological space (X, τ) whose corresponding open subset inclusion graph is complete, is characterised.

Some significant results on open subset inclusion graph of a topological space

In the recent paper R. A. Muneshwar and K. L. Bondar, introduced a graph topological structure, called open subset inclusion graph of a topological space j(t ) on a finite set X. In the present paper, we continue the study of the open subset inclusion graph of a topological space j(t ) on a finite set X regarding some important properties. It is shown that, if (X,t ) is a discrete topological space and |X|= 3, then the graph j(t ) is bipartite, regular, distance transitive, distance regular, Hamiltonian as well as vertex and edge-transitive. Also if (X,t ) is a discrete topological space and |X|= 3 then the graph j(t ) has a perfect matching and vertex connectivity and edge connectivity 2.

A subspace based subspace inclusion graph on vector space

Let $\mathscr{W}$ be a fixed $k$-dimensional subspace of an $n$-dimensi\-onal vector space $\mathscr{V}$ such that $n-k\geq1.$ In this paper, we introduce a graph structure, called the subspace based subspace inclusion graph $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}),$ where the vertex set $\mathscr{V}(\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}))$ is the collection of all subspaces $\mathscr{U}$ of $\mathscr{V}$ such that $\mathscr{U}+\mathscr{W}\neq\mathscr{V}$ and $\mathscr{U}\nsubseteq\mathscr{W},$ i.e., $\mathscr{V}(\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V}))= \{\mathscr{U}\subseteq\mathscr{V}~|~\mathscr{U}+\mathscr{W}\neq\mathscr{V}, \mathscr{U}\nsubseteq\mathscr{W}\}$ and any two distinct vertices $\mathscr{U}_{1}$ and $\mathscr{U}_{1}$ of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are adjacent if and only if either $\mathscr{U}_{1}+\mathscr{W}\subset\mathscr{U}_{2}+\mathscr{W}$ or $\mathscr{U}_{2}+\mathscr{W}\subset\mathscr{U}_{1}+\mathscr{W}.$ The diameter, girth, clique number, and chromatic number of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are studied. It is shown that two subspace based subspace inclusion graphs $\mathscr{I}_{n}^{\mathscr{W}_{1}}(\mathscr{V})$ and $\mathscr{I}_{n}^{\mathscr{W}_{2}}(\mathscr{V})$ are isomorphic if and only if $\mathscr{W}_{1}$ and $\mathscr{W}_{2}$ are isomorphic. Further, some properties of $\mathscr{I}_{n}^{\mathscr{W}}(\mathscr{V})$ are obtained when the base field is finite.

Planarity and fixing number of inclusion graph of a nilpotent group

The inclusion graph of a finite group [Formula: see text], written as [Formula: see text], is defined to be an undirected graph whose vertices are all nontrivial subgroups of [Formula: see text], and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. For a graph [Formula: see text] with vertex set [Formula: see text], a set of vertices [Formula: see text] is called a fixing set of [Formula: see text] if the only automorphism of [Formula: see text] that fixes every element in [Formula: see text] is the identity. The fixing number of [Formula: see text] is the smallest size of a fixing set of [Formula: see text]. In this paper, we determine the finite nilpotent groups whose inclusion graphs are planar. Moreover, using the technique of characteristic matrices, we characterize the fixing sets and give the exact value on the fixing number of the inclusion graphs for finite cyclic groups.

Open subset inclusion graph of a topological space

In this paper we introduce a graph topological structure, called open subset inclusion graph of a topological space on a finite set X, where the vertex set is the collection of nonempty proper open subsets of a topological space and two vertices U 1, U 2 are adjacent or U 1 ∼ U 2 or (U 1, U 2) ∈E, if either U 1 ⊂ U 2 or U 2 ⊂ U 1. The diameter, girth,connectivity, maximal independent sets, different variants of domination number, clique number and chromatic number, degree of , edge and vertex connectivity of are studied

Diameters and automorphism groups of inclusion graphs over nilpotent groups

The inclusion graph of a finite group [Formula: see text], written as [Formula: see text], is defined to be an undirected graph that its vertices are all nontrivial subgroups of [Formula: see text], and in which two distinct subgroups [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. In this paper, we determine the diameter of [Formula: see text] when [Formula: see text] is nilpotent, and characterize the independent dominating sets as well as the automorphism group of [Formula: see text].

On two conjectures on the subspace inclusion 6 graph of a vector space

The subspace inclusion graph on a vector space [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set consists of nontrivial proper subspaces of [Formula: see text] and two vertices are adjacent if one is properly contained in another. In a recent paper, Das posed the following four conjectures on the subspace inclusion graph [Formula: see text]: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular. (3) [Formula: see text] is Hamiltonian. (4) [Formula: see text] is a Cayley graph. In the present paper, we prove the first two conjectures: If [Formula: see text] is a [Formula: see text]-dimensional vector space over a finite field [Formula: see text] with [Formula: see text] elements, then: (1) The domination number of [Formula: see text] is [Formula: see text]. (2) [Formula: see text] is distance regular.

Diffusion of User Tracking Data in the Online Advertising Ecosystem

AbstractAdvertising and Analytics (A&A) companies have started collaborating more closely with one another due to the shift in the online advertising industry towards Real Time Bidding (RTB). One natural way to understand how user tracking data moves through this interconnected advertising ecosystem is by modeling it as a graph. In this paper, we introduce a novel graph representation, called anInclusiongraph, to model the impact of RTB on the diffusion of user tracking data in the advertising ecosystem. Through simulations on theInclusiongraph, we provide upper and lower estimates on the tracking information observed by A&A companies. We find that 52 A&A companies observe at least 91% of an average user’s browsing history under reasonable assumptions about information sharing within RTB auctions. We also evaluate the effectiveness of blocking strategies (e.g., AdBlock Plus), and find that major A&A companies still observe 40–90% of user impressions, depending on the blocking strategy.

Automorphism Group of the Subspace Inclusion Graph of a Vector Space

In a recent paper, Das introduced the graph $$\mathcal {I}n(\mathbb {V})$$ , called subspace inclusion graph on a finite dimensional vector space $$\mathbb {V}$$ , where the vertex set is the collection of nontrivial proper subspaces of $$\mathbb {V}$$ and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number and chromatic number of $$\mathcal {I}n(\mathbb {V})$$ when the base field is arbitrary, and he also studied some other properties of $$\mathcal {I}n(\mathbb {V})$$ when the base field is finite. At the end of the above paper, the author posed the open problem of determining the automorphisms of $$\mathcal {I}n(\mathbb {V})$$ . In this paper, we give the answer to the open problem.

Independence number of subspace inclusion graph and subspace sum graph of a vector space

ABSTRACTRecently, Das introduced the subspace inclusion graph and the subspace sum graph of a vector space , respectively, and he posed the problem of determining the independence number of both graphs when the base field of the vector space is finite. In this article, we give the answers to them.

On subspace inclusion graph of a vector space

In this paper, we continue the study of the subspace inclusion graph on a finite-dimensional vector space with dimension n where the vertex set is the collection of non-trivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. It is shown that is perfect and non-planar. Moreover, a necessary and sufficient condition is provided for to be Eulerian. For , it is shown that is bipartite, vertex and edge-transitive and has a perfect matching. We also provide exact value of the independence number and bounds on the domination number of for .

Clones of Self-Dual and Self-K-Al Functions in K-valued Logic

We give a classification of dual functions, they are m-al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and self- -al (m = k) functions in k-valued logic at k ≤ 3. And we give numerical results for clones of self-dual and self-3-al functions in 3-valued logic. In particular, the inclusion graphs of clones of self-dual and of self-3-al functions are not a lattice.

Subspace Inclusion Graph of a Vector Space

In this paper, the authors introduce a graph structure, called subspace inclusion graph ℐn(𝕍) on a finite dimensional vector space 𝕍 where the vertex set is the collection of nontrivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. The diameter, girth, clique number, and chromatic number of ℐn(𝕍) are studied. It is shown that two subspace inclusion graphs are isomorphic if and only if the base vector spaces are isomorphic. Finally, some properties of subspace inclusion graph are studied when the base field is finite.

Basic Reed–Muller Codes as Group Codes

Reed–Muller codes are one of the most well-studied families of codes; however, there are still open problems regarding their structure. Recently, a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. We recall that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. In this paper, we prove that basic Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers. This implies the same result for regular codes. Also we show how to describe the inclusion graph of basic Reed–Muller codes and radical powers via simple arithmetic equations.