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Minimal regular graphs with given girths and crossing numbers

100%
Chia G., Gan C.
Discussiones Mathematicae Graph Theory
|
2004
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tom 24
|
nr 2
223-237
EN
This paper investigates on those smallest regular graphs with given girths and having small crossing numbers.
2
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Upper Bounds on the Signed Total (K, K)-Domatic Number of Graphs

88%
Volkmann L.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 4
641-650
EN
Let G be a graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and Σx∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that Σdi=1 fi(x) ≤ k for each x ∈ V (G), is called a signed total (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed total (k, k)-dominating family on G is the signed total (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed total (k, k)- domatic number, in particular for regular graphs.
3
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A Note on the Locating-Total Domination in Graphs

88%
Miller M., Rajan R. S., Jayagopal R., Rajasingh I., Manuel P.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 3
745-754
EN
In this paper we obtain a sharp (improved) lower bound on the locating-total domination number of a graph, and show that the decision problem for the locating-total domination is NP-complete.
4
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On composition of signed graphs

88%
Shahul Hameed K., Germina K.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
|
nr 3
507-516
EN
A graph whose edges are labeled either as positive or negative is called a signed graph. In this article, we extend the notion of composition of (unsigned) graphs (also called lexicographic product) to signed graphs. We employ Kronecker product of matrices to express the adjacency matrix of this product of two signed graphs and hence find its eigenvalues when the second graph under composition is net-regular. A signed graph is said to be net-regular if every vertex has constant net-degree, namely, the difference of the number of positive and negative edges incident with a vertex. We also characterize balance in signed graph composition and have some results on the Laplacian matrices of this product.
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