Isometric classification of Sobolev spaces on graphs

M. I. Ostrovskii

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Displaying similar documents to “Isometric classification of Sobolev spaces on graphs”

Symmetries of embedded complete bipartite graphs

Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn (2014)

Fundamenta Mathematicae

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We characterize which automorphisms of an arbitrary complete bipartite graph $K_{n, m}$ can be induced by a homeomorphism of some embedding of the graph in S³.

Clopen graphs

Stefan Geschke (2013)

Fundamenta Mathematicae

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A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their...

On hereditary properties of composition graphs

Vadim E. Levit, Eugen Mandrescu (1998)

Discussiones Mathematicae Graph Theory

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The composition graph of a family of n+1 disjoint graphs $H_{i} : 0 \leq i \leq n$ is the graph H obtained by substituting the n vertices of H₀ respectively by the graphs H₁,H₂,...,Hₙ. If H has some hereditary property P, then necessarily all its factors enjoy the same property. For some sort of graphs it is sufficient that all factors $H_{i} : 0 \leq i \leq n$ have a certain common P to endow H with this P. For instance, it is known that the composition graph of a family of perfect graphs is also a perfect graph (B. Bollobas, 1978),...

Unique factorization theorem

Peter Mihók (2000)

Discussiones Mathematicae Graph Theory

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A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let ₁,₂, ...,ₙ be properties of graphs. A graph G is (₁,₂,...,ₙ)-partitionable (G has property ₁ º₂ º... ºₙ) if the vertex set V(G) of G can be partitioned into n sets V₁,V₂,..., Vₙ such that the subgraph $G [V_{i}]$ of G induced by Vi belongs to $_{i}$ ; i = 1,2,...,n. A property is said...

A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

Yury Metelsky, Kseniya Schemeleva, Frank Werner (2017)

Discussiones Mathematicae Graph Theory

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We characterize the class [...] L32 $L_{3}^{2}$ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_{3}^{2}$ in the class of threshold graphs, where n is the number of vertices of a tested graph.

The order of uniquely partitionable graphs

Izak Broere, Marietjie Frick, Peter Mihók (1997)

Discussiones Mathematicae Graph Theory

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Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition V₁,...,Vₙ of V(G) such that, for each i = 1,...,n, the subgraph of G induced by $V_{i}$ has property $_{i}$ . If a graph G has a unique (₁,...,ₙ)-partition we say it is uniquely (₁,...,ₙ)-partitionable. We establish best lower bounds for the order of uniquely (₁,...,ₙ)-partitionable graphs, for various choices of ₁,...,ₙ.

Factorizations of properties of graphs

Izak Broere, Samuel John Teboho Moagi, Peter Mihók, Roman Vasky (1999)

Discussiones Mathematicae Graph Theory

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A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs ₁,₂,...,ₙ a vertex (₁, ₂, ...,ₙ)-partition of a graph G is a partition V₁,V₂,...,Vₙ of V(G) such that for each i = 1,2,...,n the induced subgraph $G [V_{i}]$ has property $_{i}$ . The class of all graphs having a vertex (₁, ₂, ...,ₙ)-partition is denoted by ₁∘₂∘...∘ₙ. A property is said to be reducible with respect to a lattice of properties of graphs if there are n ≥ 2 properties ₁,₂,...,ₙ ∈ such that...

Bounding the Openk-Monopoly Number of Strong Product Graphs

Dorota Kuziak, Iztok Peterin, Ismael G. Yero (2018)

Discussiones Mathematicae Graph Theory

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Let G = (V, E) be a simple graph without isolated vertices and minimum degree δ, and let k ∈ 1 − ⌈δ/2⌉, . . . , ⌊δ/2⌋ be an integer. Given a set M ⊂ V, a vertex v of G is said to be k-controlled by M if [...] δM(v)≥δG(v)2+k $δ_{M} (v) \geq \frac{δ_{G} (v)}{2} + k$ , where δM(v) represents the number of neighbors of v in M and δG(v) the degree of v in G. A set M is called an open k-monopoly if every vertex v of G is k-controlled by M. The minimum cardinality of any open k-monopoly is the open k-monopoly number of G. In this...

The hull number of strong product graphs

A.P. Santhakumaran, S.V. Ullas Chandran (2011)

Discussiones Mathematicae Graph Theory

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For a connected graph G with at least two vertices and S a subset of vertices, the convex hull ${[S]}_{G}$ is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with ${[S]}_{G} = V (G)$ . Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs...

Metrically regular square of metrically regular bipartite graphs of diameter $D = 7$

Vladimír Vetchý (2018)

Archivum Mathematicum

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The present paper deals with the spectra of powers of metrically regular graphs. We prove that there is only two tables of the parameters of an association scheme so that the corresponding metrically regular bipartite graph of diameter $D = 7$ (8 distinct eigenvalues of the adjacency matrix) has the metrically regular square. The results deal with the graphs of the diameter $D < 7$ see [8], [9] and [10].

Structure of the set of all minimal total dominating functions of some classes of graphs

K. Reji Kumar, Gary MacGillivray (2010)

Discussiones Mathematicae Graph Theory

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In this paper we study some of the structural properties of the set of all minimal total dominating functions ( $_{T}$ ) of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of $_{T} (G)$ for some classes of graphs.

Reducible properties of graphs

P. Mihók, G. Semanišin (1995)

Discussiones Mathematicae Graph Theory

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Let L be the set of all hereditary and additive properties of graphs. For P₁, P₂ ∈ L, the reducible property R = P₁∘P₂ is defined as follows: G ∈ R if and only if there is a partition V(G) = V₁∪ V₂ of the vertex set of G such that $⟨ V ₁ ⟩_{G} \in P ₁$ and $⟨ V ₂ ⟩_{G} \in P ₂$ . The aim of this paper is to investigate the structure of the reducible properties of graphs with emphasis on the uniqueness of the decomposition of a reducible property into irreducible ones.

On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

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We deal with the graph operator $\bar{P o w ₂}$ defined to be the complement of the square of a graph: $\bar{P o w ₂} (G) = \bar{P o w ₂ (G)}$ . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m, n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...

2-halvable complete 4-partite graphs

Dalibor Fronček (1998)

Discussiones Mathematicae Graph Theory

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A complete 4-partite graph $K_{m ₁, m ₂, m ₃, m ₄}$ is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with at most one odd part all d-halvable graphs are known. In the class of biregular graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with four odd parts (i.e., the graphs $K_{m, m, m, n}$ and $K_{m, m, n, n}$ ) all d-halvable graphs are known as well, except for the graphs $K_{m, m, n, n}$ when d = 2 and n ≠ m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with three...

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

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A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

Edge-connectivity of strong products of graphs

Bostjan Bresar, Simon Spacapan (2007)

Discussiones Mathematicae Graph Theory

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The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_{i} = y_{i}$ or $x_{i} y_{i} \in E (G_{i})$ . In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.

Growth in ${SL}_{3} (ℤ / p ℤ)$

H. A. Helfgott (2011)

Journal of the European Mathematical Society

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Radio numbers for generalized prism graphs

Paul Martinez, Juan Ortiz, Maggy Tomova, Cindy Wyels (2011)

Discussiones Mathematicae Graph Theory

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A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted $Z_{n, s}$ , s ≥ 1, n ≥ s, have vertex set (i,j) | i = 1,2 and j = 1,...,n and edge set ((i,j),(i,j ±1)) ∪ ((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋. In this paper we determine...

Stable sets for $(P ₆, K_{2, 3})$ -free graphs

Raffaele Mosca (2012)

Discussiones Mathematicae Graph Theory

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The Maximum Stable Set (MS) problem is a well known NP-hard problem. However different graph classes for which MS can be efficiently solved have been detected and the augmenting graph technique seems to be a fruitful tool to this aim. In this paper we apply a recent characterization of minimal augmenting graphs [22] to prove that MS can be solved for $(P ₆, K_{2, 3})$ -free graphs in polynomial time, extending some known results.

Generalized chromatic numbers and additive hereditary properties of graphs

Izak Broere, Samantha Dorfling, Elizabeth Jonck (2002)

Discussiones Mathematicae Graph Theory

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An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let and be additive hereditary properties of graphs. The generalized chromatic number $χ_{} ()$ is defined as follows: $χ_{} () = n$ iff ⊆ ⁿ but $⊊^{n - 1}$ . We investigate the generalized chromatic numbers of the well-known properties of graphs ₖ, ₖ, ₖ, ₖ and ₖ.

Finite groups whose character degree graphs coincide with their prime graphs

Temha Erkoç, Utku Yilmaztürk, İsmail Ş. Güloğlu (2018)

Czechoslovak Mathematical Journal

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In the literature, there are several graphs related to a finite group $G$ . Two of them are the character degree graph, denoted by $Δ (G)$ , and the prime graph, $Γ (G)$ . In this paper we classify all finite groups whose character degree graphs are disconnected and coincide with their prime graphs. As a corollary, we find all finite groups whose character degree graphs are square and coincide with their prime graphs.

Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Jozef Bucko, Jaroslav Ivančo (1999)

Discussiones Mathematicae Graph Theory

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Let ₁, ₂ be graph properties. A vertex (₁,₂)-partition of a graph G is a partition V₁,V₂ of V(G) such that for i = 1,2 the induced subgraph $G [V_{i}]$ has the property $_{i}$ . A property ℜ = ₁∘₂ is defined to be the set of all graphs having a vertex (₁,₂)-partition. A graph G ∈ ₁∘₂ is said to be uniquely (₁,₂)-partitionable if G has exactly one vertex (₁,₂)-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a...