Unique Perfect Matching Research Articles

Bipartite unicyclic graphs with a unique perfect matching having the smallest positive eigenvalue equal to

The smallest positive eigenvalue τ ( G ) of a simple graph G is the smallest positive eigenvalue of its adjacency matrix A ( G ) . In [F. J. Zhang and A. Chang, Acyclic molecules with greatest HOMO-LUMO separation, Discrete Applied Mathematics, 98:165–171, (1999).], the authors characterized all nonsingular trees with τ equal to 2 − 1 . We consider the same problem for bipartite unicyclic graphs with a unique perfect matching. Let U be the class of all connected bipartite unicyclic graphs with a unique perfect matching. In this article, we characterize all graphs U in U with the property that τ ( U ) = 2 − 1 . Further, we show that the largest limit point of the smallest positive eigenvalues of graphs in U is 2 − 1 , whereas the smallest limit point is 0.

Inverse of \(\alpha\)-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.

Removable and forced subgraphs of graphs

A connected graph G is H-removable if H is a subgraph of G, G has at least |V(H)|+2 vertices and G−V(H′) has a perfect matching for every subgraph H′ of G isomorphic to H. So a connected graph is kK2-removable if and only if it is k-extendable, where kK2 is a matching of size k. Further G is an H-forced graph if G−V(H′) has a unique perfect matching for every subgraph H′ of G isomorphic to H. In this paper, we first characterize kK2-forced graphs for k≥2 as K2k+2 and Kk+1,k+1 by using randomly matchable graphs. Let Pi denote a path with i vertices. We show that P3- and P4-removable graphs are factor-critical and 1-extendable respectively. By using ear decomposition, we obtain that a connected graph G is P3-forced if and only if G is K5 or C2k+1 (k≥2). Finally we prove that a connected graph G is P4-forced if and only if G is isomorphic to K6, K3,3, C2k (k≥3), the Petersen graph, a uniform theta graph Θ3n (n≥3) or C12+.

On unimodular graphs with a unique perfect matching

A graph is called unimodular if its adjacency matrix has determinant ±1. This article provides a necessary and sufficient condition for a simple connected graph with a unique perfect matching to be unimodular. In particular, we give a complete characterization of bicyclic unimodular graphs with a unique perfect matching. Moreover, the possible values of the determinant of the adjacency matrix of unicyclic, bicyclic, and tricyclic graphs with a unique perfect matching are also provided in this article. For non-bipartite unicyclic graphs with a unique perfect matching, we address the problem of when the inverse of the corresponding adjacency matrix is diagonally similar to a non-negative matrix. A pseudo-unimodular graph is a singular graph whose product of non-zero eigenvalues of the corresponding adjacency matrix is ±1. We supply a necessary and sufficient condition for a singular graph to be pseudo-unimodular.

On the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching II

Let G be a simple graph with the adjacency matrix A ( G ) . Let τ ( G ) denote the smallest positive eigenvalue of A ( G ) . In 1990, Pavlíková and Kr c ˘ -Jediný proved that among all nonsingular trees on n = 2m vertices, the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. We consider the problem for unicyclic graphs. Let U denote the class of all connected bipartite unicyclic graphs with a unique perfect matching, and for each m ≥ 3 , let U n be the subclass of U with graphs on n = 2m vertices. We first obtain the classes of unicyclic graphs U in U such that τ ( U ) ≤ 2 − 1 . We then find the unique graph U o n (resp. U e n ) having the maximum τ value among all graphs in U n when m is odd (resp. when m is even). Finally, we prove that U o 6 (the graph obtained from a cycle of order 4, by adding two pendants to two adjacent vertices) is the graph with maximum τ value among all graphs in U . As a consequence, we obtain a sharp upper bound for τ ( U ) when U ∈ U .

Some tight bounds on the minimum and maximum forcing numbers of graphs

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$. We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0\leq k\leq n-1$, and characterize corresponding extremal graphs as well. Hence we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. Further, we obtain a new upper bound of $F(G)$. For bipartite graphs $G$, Che and Chen (2013) obtained that $f(G)=n-1$ if and only if $G$ is complete bipartite graph $K_{n,n}$. We completely characterize all bipartite graphs $G$ with $f(G)= n-2$.

On the smallest positive eigenvalue of bipartite unicyclic graphs with a unique perfect matching

Let G be a simple graph with the adjacency matrix A(G). Let τ(G) denote the smallest positive eigenvalue of A(G). In 1985, Godsil proved that in the class of all nonsingular trees on n=2m vertices, the path on 2m vertices has the minimum τ value. We consider the same problem for unicyclic graphs. Let Un be the class of connected bipartite unicyclic graphs on n=2m vertices, with a unique perfect matching. Let us take the path P2m=[1,2,…,m,m+1,…,2m]. If m is even, let Ue be the graph obtained from P2m by adding an edge between the vertices m−2 and m+3. If m is odd, let Uo be the graph obtained from the path P2m by adding an edge between the vertices m−3 and m+2. We show that the unique graph Ue (resp. Uo) has the minimum τ value among all the graphs in U2m when m is even (resp. when m is odd). In 1990, Pavlíková and Krč-Jediný proved that among all nonsingular trees on n=2m vertices the comb graph (obtained by taking a path on m vertices and adding a new pendant vertex to every vertex of the path) has the maximum τ value. Thus, it can be observed that there are only three trees with the smallest positive eigenvalue greater than 12. In this article, we characterize all bipartite unicyclic graphs G with a unique perfect matching such that τ(G)<12 (ultimately, we obtain only two bipartite unicyclic graphs with a unique perfect matching such that τ(G)>12).

Unicyclic 3-colored digraphs with bicyclic inverses

The class of unicyclic $3$-colored digraphs with the cycle weight $\pm\mathrm{i}$ and with a unique perfect matching, denoted by $\mathcal{U}_g$, is considered in this article. Kalita \&amp; Sarma [On the inverse of unicyclic 3-coloured digraphs, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2021.1948956] introduced the notion of inverse of $3$-colored digraphs. They characterized the unicyclic $3$-colored digraphs in $\mathcal{U}_g$ possessing unicyclic inverses. This article provides a complete characterization of the unicyclic $3$-colored digraphs in $\mathcal{U}_g$ possessing bicyclic inverses.

A q-analogue of the bipartite distance matrix of a nonsingular tree

All graphs considered here are simple and finite. Let G be a labeled, connected, bipartite graph and let (L,R) be a vertex bipartition of G. The bipartite adjacency matrix A of G is the submatrix of its adjacency matrix determined by the rows corresponding to L and the columns corresponding to R, see [6]. In a similar way, one can consider the bipartite distance matrix of G. It is the submatrix of the usual distance matrix D of G determined by the rows corresponding to L and the columns corresponding to R. Let G=T be a tree with a unique perfect matching. Very recently, in [2], it was shown that this matrix possesses many interesting properties and has a very deep combinatorial relation with the tree structure. One of those results, is the elegant description of the determinant of the bipartite distance matrix of T by a combinatorial sum over the collection of all alternating paths. Another result is reminiscent of the well known result of Graham and Pollack, which tells that the determinant of the usual distance matrix a tree of order 2p is a multiple of 22p−2. It was shown that the determinant of the bipartite distance matrix of a tree of order 2p with a unique perfect matching is a multiple of 2p−1.There are many studies of distance-like matrices available in the literature. In this article, we consider two of them. The first one is the q-distance matrix which is a generalization of the usual distance matrix. We show that the determinant of the q-bipartite distance matrix of a tree with a unique perfect matching can still be expressed as a combinatorial sum over the set of alternating paths with respect to a suitable sequence. The result of [2] follows as a corollary when q=1.The second distance-like matrix we study is the exponential distance matrix, which has a very simple expression for the determinant. We show that an extremely similar result hold here too. This is unexpected, as our matrix is of half the order.Recall that, the determinant of the usual distance matrix of a tree is just dependent on the number of vertices and is independent of the tree structure. In [2], it was shown that the determinant of the bipartite distance matrix of a tree with a unique perfect matching was dependent on the tree structure and on some smaller classes it was independent of the tree structure. We prove similar results for the q-bipartite distance matrix.However, as another surprise, we show that the determinant of the exponential bipartite distance matrix of a tree with a unique perfect matching is independent of the tree structure.

Distance spectrum, 1-factor and vertex-disjoint cycles

Motivated by recent progresses on study of factors and spectral extremal graph theory, in this paper, we focus on distance spectral extrema of k-factors in bipartite graphs for small k. First of all, we aim to investigate the existence of 1-factor in a bipartite graph with given minimum degree in terms of its distance spectral radius, which generalize and improve the previous result in [37]. Then we determine the extremal graph attaining the minimum distance spectral radius among all bipartite graphs with a unique perfect matching. Notice that a 2-factor consists of some vertex-disjoint cycles. As a beginning, we naturally consider some propositions of vertex-disjoint cycles and give a sufficient condition for the existence of two vertex-disjoint cycles in a bipartite graph with respect to the distance spectral radius.

On unicyclic non-bipartite graphs with tricyclic inverses

The class of unicyclic non-bipartite graphs with unique perfect matching, denoted by , is considered in this article. This article provides a complete characterization of unicyclic graphs in which possess tricyclic inverses.

Continuous forcing spectrum of regular hexagonal polyhexes

For any perfect matching M of a graph G, the forcing number (resp. anti-forcing number) of M is the smallest cardinality of an edge subset S⊆M (resp. S⊆E(G)∖M) such that the graph G−V(S) (resp. G−S) has a unique perfect matching. The forcing spectrum of G is the set of forcing numbers of all perfect matchings of G. Afshani et al. [2] proved that any finite set of positive integers can be the forcing spectrum of a planar bipartite graph. A polyhex is a 2-connected plane bipartite graph whose interior faces are regular hexagons. In this paper, we give the minimum forcing number and anti-forcing number of a polyhex with a 3-divisible perfect matching. Further, we prove that the forcing spectrum of any regular hexagonal polyhex is continuous, i.e. an integer interval, by applying some changes on prolate triangle polyhexes to obtain a sequence of perfect matchings of the graph.

Relations between global forcing number and maximum anti-forcing number of a graph

The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For a perfect matching M of G, the minimal cardinality of an edge subset S⊆E(G)∖M such that G−S has a unique perfect matching is called the anti-forcing number of M. The maximum anti-forcing number of G among all perfect matchings is denoted by Af(G). It is known that the maximum anti-forcing number of a hexagonal system equals the famous Fries number.For a bipartite graph G, we show that gf(G)≥Af(G). Next we extend such result to Birkhoff–von Neumann graphs, whose perfect matching polytopes are characterized solely by nonnegativity and degree constraints, by revealing an odd dumbbell of non-bipartite graphs with a unique perfect matching and minimum degree at least two. Finally, we obtain tight upper and lower bounds on gf(G)−Af(G). For a connected bipartite graph G with 2n vertices, 0≤gf(G)−Af(G)≤12(n−1)(n−2). For non-bipartite case, we have −Occ(G)≤gf(G)−Af(G)≤(n−1)(n−2) by introducing a new nonnegative parameter Occ(G).

The bipartite distance matrix of a nonsingular tree

Let G be a labeled, connected, bipartite graph with the bi-partition (L={l1,…,lk},R={r1,…,rp}) of the vertex set V. Let D be the usual distance matrix of G, where rows and columns are indexed by l1,…,lk,r1,…,rp. For X,Y⊆V, let us define DG[X,Y] as the submatrix of D induced by the rows indexed in X and columns indexed in Y. Let us call DG[L,R] the bipartite distance matrix of G. If G has a unique perfect matching, then k=p and we assume that the bi-partition is canonical, that is, [li,ri] are matching edges. For a nonsingular tree T, let us denote the bipartite distance matrix of T by B(T).We observe that det⁡B(T) is always a multiple of 2p−1. This is similar to the well known result of Graham and Pollak (1971) [1] which tells that the determinant of the usual distance matrix D is a multiple of 2n−2. Call the number bd(T):=det⁡B(T)/(−2)p−1 the bipartite distance index of T. We supply a recursive formula to compute this index. We show that this index satisfies an interesting inclusion-exclusion type of principle at any matching edge of the tree. Even more interestingly, we show that the index is completely characterized by the structure of T via what we call the f-alternating sums, that is, the sum f(T):=∑[d(u)−2][d(v)−2]S|Puv|/2, where the sum is taken over all u-v-alternating paths Puv, and S is the sequence (1,1,3,3,5,5,…).A well known result by Graham, Hoffman and Hosoya (1977) [2] is that the determinant of the distance matrix of a graph only depends on the blocks and is independent of how they are assembled. Such a result does not hold true for B(T). However, we identify some basic elements and a merging operation and show that each of the trees that can be constructed from a given set of elements, sequentially using this operation, have the same det⁡B(T), independent of the order in which the sequence is followed. For the class of trees that can be obtained in this way, we give a surprisingly simple way to evaluate the determinant of B(T).

On the inverse of unicyclic 3-coloured digraphs

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight and with a unique perfect matching, denoted by , is considered in this article. We characterize the 3-coloured digraphs in whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight satisfying the strong reciprocal eigenvalue property.

Characterizing the fullerene graphs with the minimum forcing number 3

The minimum forcing number of a graph G is the smallest number of edges simultaneously contained in a unique perfect matching of G. Zhang et al. (2010) claimed that the minimum forcing number of any fullerene graph was bounded below by 3. However, we find that there exists exactly one exceptional fullerene F24 with the minimum forcing number 2. In this paper, we characterize all fullerenes with the minimum forcing number 3 by a construction approach. This also solves an open problem proposed by Zhang et al. We also find that except for F24, all fullerenes with anti-forcing number 4 have the minimum forcing number 3. In particular, the nanotube fullerenes of type (4,2) are such fullerenes.

Classes of nonbipartite graphs with reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A(G) is nonsingular and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity, for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).

Inverses of non-bipartite unicyclic graphs with a unique perfect matching

The class of non-bipartite unicyclic graphs with a unique perfect matching, denoted by , is considered in this article. This article describes the entries of the inverse of the adjacency matrix of a graph in . It is proved that the inverse graph of a graph in is always non-bipartite. In this article, we characterize the graphs in whose inverses are mixed graphs. Among all such graphs in we identify those graphs whose inverses are quasi-bipartite. Furthermore, characterizations of unicyclic graphs in possessing unicyclic and bicyclic inverses are also provided in this article.

Strongly self-inverse weighted graphs

Let G be a connected, bipartite graph. Let Gw denote the weighted graph obtained from G by assigning weights to its edges using the positive weight function w : E(G) ! (0;1). In this article we consider a class Hnmc of bipartite graphswith unique perfect matchings and the family WG of weight functions with weight 1 on the matching edges, and characterize all pairs G in Hnmc and w in WG such that Gw is strongly self-inverse.

On the equality of the induced matching number and the uniquely restricted matching number for subcubic graphs

For a matching M in a graph G, let G(M) be the subgraph of G induced by the vertices of G that are incident with an edge in M. The matching M is induced, if G(M) is 1-regular, and M is uniquely restricted, if M is the unique perfect matching of G(M). The induced matching number νs(G) of G is the largest size of an induced matching in G, and the uniquely restricted matching number νur(G) of G is the largest size of a uniquely restricted matching in G.Golumbic et al. (2001) [4] posed the problem to characterize the graphs G with νs(G)=νur(G). We give a complete characterization of the 2-connected subcubic graphs G of sufficiently large order with νs(G)=νur(G). As a consequence, we are able to show that the subcubic graphs G with νs(G)=νur(G) can be recognized in polynomial time.