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1
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Restrained domination in unicyclic graphs

100%
Hattingh J., Joubert E., Loizeaux M., Plummer A., van der Merwe L.
Discussiones Mathematicae Graph Theory
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2009
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tom 29
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nr 1
71-86
EN
Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by $γ_r(G)$, is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then $γ_r(U) ≥ ⎡n/3⎤$, and provide a characterization of graphs achieving this bound.
2
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On non-z(mod k) dominating sets

80%
Caro Y., Jacobson M.
Discussiones Mathematicae Graph Theory
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2003
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tom 23
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nr 1
189-199
EN
For a graph G, a positive integer k, k ≥ 2, and a non-negative integer with z < k and z ≠ 1, a subset D of the vertex set V(G) is said to be a non-z (mod k) dominating set if D is a dominating set and for all x ∈ V(G), |N[x]∩D| ≢ z (mod k).For the case k = 2 and z = 0, it has been shown that these sets exist for all graphs. The problem for k ≥ 3 is unknown (the existence for even values of k and z = 0 follows from the k = 2 case.) It is the purpose of this paper to show that for k ≥ 3 and with z < k and z ≠ 1, that a non-z(mod k) dominating set exist for all trees. Also, it will be shown that for k ≥ 4, z ≥ 1, 2 or 3 that any unicyclic graph contains a non-z(mod k) dominating set. We also give a few special cases of other families of graphs for which these dominating sets must exist.
3
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The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic

80%
Wang Y., Fan Y.-Z., Li X.-X., Zhang F.-F.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
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nr 2
249-260
EN
A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20.
4
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Extremal Unicyclic Graphs With Minimal Distance Spectral Radius

80%
Lu H., Luo J., Zhu Z.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 4
735-749
EN
The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) \ Cn.
5
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Order unicyclic graphs according to spectral radius of unoriented laplacian matrix

80%
Fan Y.-Z., Wu S.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 3
487-499
EN
The spectral radius of a graph is defined by that of its unoriented Laplacian matrix. In this paper, we determine the unicyclic graphs respectively with the third and the fourth largest spectral radius among all unicyclic graphs of given order.
6
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The inertia of unicyclic graphs and bicyclic graphs

70%
Liu Y.
Discussiones Mathematicae - General Algebra and Applications
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2013
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tom 33
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nr 1
109-115
EN
Let G be a graph with n vertices and ν(G) be the matching number of G. The inertia of a graph G, In(G) = (n₊,n₋,n₀) is an integer triple specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix A(G), respectively. Let η(G) = n₀ denote the nullity of G (the multiplicity of the eigenvalue zero of G). It is well known that if G is a tree, then η(G) = n - 2ν(G). Guo et al. [Ji-Ming Guo, Weigen Yan and Yeong-Nan Yeh. On the nullity and the matching number of unicyclic graphs, Linear Algebra and its Applications, 431 (2009), 1293-1301.] proved if G is a unicyclic graph, then η(G) equals n - 2ν(G) - 1, n-2ν(G) or n - 2ν(G) + 2. Barrett et al. determined the inertia sets for trees and graphs with cut vertices. In this paper, we give the nullity of bicyclic graphs 𝓑ₙ⁺⁺. Furthermore, we determine the inertia set in unicyclic graphs and 𝓑ₙ⁺⁺, respectively.
7
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Nonsingular unicyclic mixed graphs with at most three eigenvalues greater than two

70%
Gong S.-C., Fan Y.-Z.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
69-82
EN
This paper determines all nonsingular unicyclic mixed graphs on at least nine vertices with at most three Laplacian eigenvalues greater than two.
8
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Solutions of Some L(2, 1)-Coloring Related Open Problems

61%
Mandal N., Panigrahi P.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 2
279-297
EN
An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f : V (G) → Z+ ∪ {0} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of the graph is called a span coloring. For an L(2, 1)-coloring f of a graph G with span k, an integer h is a hole in f if h ∈ (0, k) and there is no vertex v in G such that f(v) = h. A no-hole coloring is an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is irreducible if color of none of the vertices in the graph can be decreased to yield another L(2, 1)-coloring of the same graph. A graph G is inh-colorable if there exists an irreducible no-hole coloring of G. Most of the results obtained in this paper are answers to some problems asked by Laskar et al. [5]. These problems are mainly about relationship between the span and maximum no-hole span of a graph, lower inh-span and upper inh-span of a graph, and the maximum number of holes and minimum number of holes in a span coloring of a graph. We also give some sufficient conditions for a tree and an unicyclic graph to have inh-span Δ + 1.
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