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A Maximum Resonant Set of Polyomino Graphs

100%
Zhang H., Zhou X.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 2
323-337
EN
A polyomino graph P is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. A dimer covering of P corresponds to a perfect matching. Different dimer coverings can interact via an alternating cycle (or square) with respect to them. A set of disjoint squares of P is a resonant set if P has a perfect matching M so that each one of those squares is M-alternating. In this paper, we show that if K is a maximum resonant set of P, then P − K has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to the cardinality of a maximum resonant set. This confirms a conjecture of Xu et al. [26]. We also show that if K is a maximal alternating set of P, then P − K has a unique perfect matching.
2
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Per-Spectral Characterizations Of Some Bipartite Graphs

100%
Wu T., Zhang H.
Discussiones Mathematicae Graph Theory
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2017
|
tom 37
|
nr 4
935-951
EN
A graph is said to be characterized by its permanental spectrum if there is no other non-isomorphic graph with the same permanental spectrum. In this paper, we investigate when a complete bipartite graph Kp,p with some edges deleted is determined by its permanental spectrum. We first prove that a graph obtained from Kp,p by deleting all edges of a star K1,l, provided l < p, is determined by its permanental spectrum. Furthermore, we show that all graphs with a perfect matching obtained from Kp,p by removing five or fewer edges are determined by their permanental spectra.
3
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Sharp Upper Bounds on the Clar Number of Fullerene Graphs

100%
Gao Y., Zhang H.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
|
nr 1
155-163
EN
The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30.
4
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A Note on the Permanental Roots of Bipartite Graphs

81%
Zhang H., Liu S., Li W.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
|
nr 1
49-56
EN
It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imaginary axes. Furthermore, we prove that any graph has no negative real permanental root, and any graph containing at least one edge has complex permanental roots.
5
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Extremal Matching Energy of Complements of Trees

81%
Wu T., Yan W., Zhang H.
Discussiones Mathematicae Graph Theory
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2016
|
tom 36
|
nr 3
505-521
EN
Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have the minimum matching energy for p = 1, 2, . . . , [n/2]. When we restrict our consideration to all trees with a perfect matching, we determine the trees whose complements have the second-maximal matching energy.
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