Intersection graph of gamma sets in the total graph

T. Tamizh Chelvam; T. Asir

Displaying similar documents to “Intersection graph of gamma sets in the total graph”

Iterated neighborhood graphs

Martin Sonntag, Hanns-Martin Teichert (2012)

Discussiones Mathematicae Graph Theory

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The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph $(V, E_{N})$ where $E_{N}$ = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph $N^{k} (G) : = N (N (. . . N (G)))$ of G. In particular we investigate conditions for G and k such that $N^{k} (G)$ becomes a complete graph.

Remarks on partially square graphs, hamiltonicity and circumference

Hamamache Kheddouci (2001)

Discussiones Mathematicae Graph Theory

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Given a graph G, its partially square graph G* is a graph obtained by adding an edge (u,v) for each pair u, v of vertices of G at distance 2 whenever the vertices u and v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In the case where G is a claw-free graph, G* is equal to G². We define $σ ° ₜ = m i n \sum_{x \in S} d_{G} (x) : S i s a n i n d e p e n d e n t s e t i n G * a n d | S | = t$ . We give for hamiltonicity and circumference new sufficient conditions depending on σ° and we improve some known results.

Potentially H-bigraphic sequences

Michael Ferrara, Michael Jacobson, John Schmitt, Mark Siggers (2009)

Discussiones Mathematicae Graph Theory

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We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of...

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

When a line graph associated to annihilating-ideal graph of a lattice is planar or projective

Atossa Parsapour, Khadijeh Ahmad Javaheri (2018)

Czechoslovak Mathematical Journal

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Let $(L, \land, \lor)$ be a finite lattice with a least element 0. $𝔸 G (L)$ is an annihilating-ideal graph of $L$ in which the vertex set is the set of all nontrivial ideals of $L$ , and two distinct vertices $I$ and $J$ are adjacent if and only if $I \land J = 0$ . We completely characterize all finite lattices $L$ whose line graph associated to an annihilating-ideal graph, denoted by $𝔏 (𝔸 G (L))$ , is a planar or projective graph.

Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

Éric Sopena (2012)

Discussiones Mathematicae Graph Theory

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The oriented chromatic number of an oriented graph $^{\to} G$ is the minimum order of an oriented graph $^{\to} H$ such that $^{\to} G$ admits a homomorphism to $^{\to} H$ . The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the minimum order of an oriented graph $^{\to} U$ such that every orientation $^{\to} G$ of G admits a homomorphism to $^{\to} U$ . We give...

The spectral determinations of the connected multicone graphs $K_{w} ▽ m P_{17}$ and $K_{w} ▽ m S$

Ali Zeydi Abdian, S. Morteza Mirafzal (2018)

Czechoslovak Mathematical Journal

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Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let $K_{w}$ denote a complete graph on $w$ vertices, and let $m$ be a positive integer number. In A. Z. Abdian (2016)...

On the order of certain close to regular graphs without a matching of given size

Sabine Klinkenberg, Lutz Volkmann (2007)

Czechoslovak Mathematical Journal

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A graph $G$ is a ${d, d + k}$ -graph, if one vertex has degree $d + k$ and the remaining vertices of $G$ have degree $d$ . In the special case of $k = 0$ , the graph $G$ is $d$ -regular. Let $k, p \geq 0$ and $d, n \geq 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected ${d, d + k}$ -graph of order $n$ without a matching $M$ of size $2 | M | = n - p$ , then we show in this paper the following: If $d = 2$ , then $k \geq 2 (p + 2)$ and (i) $n \geq k + p + 6$ . If $d \geq 3$ is odd and $t$ an integer with $1 \leq t \leq p + 2$ , then (ii) $n \geq d + k + 1$ for $k \geq d (p + 2)$ , (iii) $n \geq d (p + 3) + 2 t + 1$ for $d (p + 2 - t) + t \leq k \leq d (p + 3 - t) + t - 3$ , (iv) $n \geq d (p + 3) + 2 p + 7$ for $k \leq p$ . If $d \geq 4$ is even, then (v) $n \geq d + k + 2 - η$ for $k \geq d (p + 3) + p + 4 + η$ , (vi) $n \geq d + k + p + 2 - 2 t = d (p + 4) + p + 6$ for $k = d (p + 3) + 4 + 2 t$ and $p \geq 1$ ,...

The chromatic equivalence class of graph $\bar{B_{n - 6, 1, 2}}$

Jianfeng Wang, Qiongxiang Huang, Chengfu Ye, Ruying Liu (2008)

Discussiones Mathematicae Graph Theory

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By h(G,x) and P(G,λ) we denote the adjoint polynomial and the chromatic polynomial of graph G, respectively. A new invariant of graph G, which is the fourth character R₄(G), is given in this paper. Using the properties of the adjoint polynomials, the adjoint equivalence class of graph $B_{n - 6, 1, 2}$ is determined, which can be regarded as the continuance of the paper written by Wang et al. [J. Wang, R. Liu, C. Ye and Q. Huang, A complete solution to the chromatic equivalence class of graph $\bar{B_{n - 7, 1, 3}}$ , Discrete...

On the intersection graph of a finite group

Hossein Shahsavari, Behrooz Khosravi (2017)

Czechoslovak Mathematical Journal

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For a finite group $G$ , the intersection graph of $G$ which is denoted by $Γ (G)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H \cap K \neq 1$ . In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of $Aut (Γ (G))$ .

On the total k-domination number of graphs

Adel P. Kazemi (2012)

Discussiones Mathematicae Graph Theory

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Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number $γ_{\times k} (G)$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, $| N_{G} [v] \cap S | \geq k$ . Also the total k-domination number $γ_{\times k, t} (G)$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, $| N_{G} (v) \cap S | \geq k$ . The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for...

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

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The domination subdivision number $s d_{γ} (G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

A note on periodicity of the 2-distance operator

Bohdan Zelinka (2000)

Discussiones Mathematicae Graph Theory

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The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_{f}$ of all finite undirected graphs. If G is a graph from $C_{f}$ , then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.

Nonempty intersection of longest paths in a graph with a small matching number

Fuyuan Chen (2015)

Czechoslovak Mathematical Journal

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A maximum matching of a graph $G$ is a matching of $G$ with the largest number of edges. The matching number of a graph $G$ , denoted by $α^{'} (G)$ , is the number of edges in a maximum matching of $G$ . In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Although this conjecture has been disproved, finding some nice classes of graphs that support this conjecture is still very meaningful and interesting. In this short note, we prove that Gallai’s conjecture...