Wiener and vertex PI indices of the strong product of graphs

K. Pattabiraman; P. Paulraja

Displaying similar documents to “Wiener and vertex PI indices of the strong product of graphs”

The Wiener number of powers of the Mycielskian

Rangaswami Balakrishnan, S. Francis Raj (2010)

Discussiones Mathematicae Graph Theory

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The Wiener number of a graph G is defined as $1 / 2 \sum_{u, v \in V (G)} d (u, v)$ , d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, $W (μ (S ₙ^{k})) \leq W (μ (T ₙ^{k})) \leq W (μ (P ₙ^{k}))$ , where Sₙ, Tₙ and Pₙ denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of $μ (G^{k})$ .

Wiener index of graphs with fixed number of pendant or cut-vertices

Dinesh Pandey, Kamal Lochan Patra (2022)

Czechoslovak Mathematical Journal

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The Wiener index of a connected graph is defined as the sum of the distances between all unordered pairs of its vertices. We characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut-vertices.

Domination Subdivision Numbers

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi, David P. Jacobs, James Knisely, Lucas C. van der Merwe (2001)

Discussiones Mathematicae Graph Theory

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A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number $s d_{γ} (G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam conjectured that $1 \leq s d_{γ} (G) \leq 3$ for any graph G. We give a counterexample to this conjecture. On the other hand,...

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...

The extremal irregularity of connected graphs with given number of pendant vertices

Xiaoqian Liu, Xiaodan Chen, Junli Hu, Qiuyun Zhu (2022)

Czechoslovak Mathematical Journal

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The irregularity of a graph $G = (V, E)$ is defined as the sum of imbalances $| d_{u} - d_{v} |$ over all edges $u v \in E$ , where $d_{u}$ denotes the degree of the vertex $u$ in $G$ . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with $n$ vertices and $p$ pendant vertices ( $1 \leq p \leq n - 1$ ), and characterize the corresponding extremal graphs.

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 $p \geq 1$ ,...

Roman bondage in graphs

Nader Jafari Rad, Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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A Roman dominating function on a graph G is a function f:V(G) → 0,1,2 satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f (V (G)) = \sum_{u \in V (G)} f (u)$ . The Roman domination number, $γ_{R} (G)$ , of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_{R} (G)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E’ ⊆ E(G)...

Full domination in graphs

Robert C. Brigham, Gary Chartrand, Ronald D. Dutton, Ping Zhang (2001)

Discussiones Mathematicae Graph Theory

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For each vertex v in a graph G, let there be associated a subgraph $H_{v}$ of G. The vertex v is said to dominate $H_{v}$ as well as dominate each vertex and edge of $H_{v}$ . A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number $γ_{F H} (G)$ . A full dominating set of G of cardinality $γ_{F H} (G)$ is called a $γ_{F H}$ -set of G. We study three types of full domination in...

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...

The geodetic number of strong product graphs

A.P. Santhakumaran, S.V. Ullas Chandran (2010)

Discussiones Mathematicae Graph Theory

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For two vertices u and v of a connected graph G, the set $I_{G} [u, v]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets $I_{G} [u, v]$ for u,v ∈ S is denoted by $I_{G} [S]$ . A set S ⊆ V(G) is a geodetic set if $I_{G} [S] = V (G)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.

The vertex detour hull number of a graph

A.P. Santhakumaran, S.V. Ullas Chandran (2012)

Discussiones Mathematicae Graph Theory

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For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_{D} [S] = ⋃_{x, y \in S} I_{D} [x, y]$ . A set S of vertices is a detour convex set if $I_{D} [S] = S$ . The detour convex hull ${[S]}_{D}$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of...

On locating-domination in graphs

Mustapha Chellali, Malika Mimouni, Peter J. Slater (2010)

Discussiones Mathematicae Graph Theory

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A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number $γ_{L} (G)$ is the minimum cardinality of a LDS of G, and the upper locating-domination number, $Γ_{L} (G)$ is the maximum cardinality of a minimal LDS of G. We present different bounds on $Γ_{L} (G)$ and $γ_{L} (G)$ .

Edge-sum distinguishing labeling

Jan Bok, Nikola Jedličková (2021)

Commentationes Mathematicae Universitatis Carolinae

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We study edge-sum distinguishing labeling, a type of labeling recently introduced by Z. Tuza (2017) in context of labeling games. An ESD labeling of an $n$ -vertex graph $G$ is an injective mapping of integers $1$ to $l$ to its vertices such that for every edge, the sum of the integers on its endpoints is unique. If $l$ equals to $n$ , we speak about a canonical ESD labeling. We focus primarily on structural properties of this labeling and show for several classes of graphs if they have or do not...

Characterization by intersection graph of some families of finite nonsimple groups

Hossein Shahsavari, Behrooz Khosravi (2021)

Czechoslovak Mathematical Journal

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For a finite group $G$ , $Γ (G)$ , the intersection graph of $G$ , is a simple graph whose 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 nonsimple groups whose intersection graphs have a leaf and also we discuss the characterizability of them using their intersection graphs.

Stronger bounds for generalized degrees and Menger path systems

R.J. Faudree, Zs. Tuza (1995)

Discussiones Mathematicae Graph Theory

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For positive integers d and m, let $P_{d, m} (G)$ denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that $P_{d, m} (G)$ is satisfied have been investigated. In particular, it has been shown,...