The Graphs Whose Permanental Polynomials Are Symmetric

Wei Li

Displaying similar documents to “The Graphs Whose Permanental Polynomials Are Symmetric”

Counterexample to a conjecture on the structure of bipartite partitionable graphs

Richard G. Gibson, Christina M. Mynhardt (2007)

Discussiones Mathematicae Graph Theory

Similarity:

A graph G is called a prism fixer if γ(G×K₂) = γ(G), where γ(G) denotes the domination number of G. A symmetric γ-set of G is a minimum dominating set D which admits a partition D = D₁∪ D₂ such that $V (G) - N [D_{i}] = D_{j}$ , i,j = 1,2, i ≠ j. It is known that G is a prism fixer if and only if G has a symmetric γ-set. Hartnell and Rall [On dominating the Cartesian product of a graph and K₂, Discuss. Math. Graph Theory 24 (2004), 389-402] conjectured that if G is a connected, bipartite graph such that V(G) can...

On a class of polynomials associated with the cliques in a graph and its applications.

Farrell, E.J. (1989)

International Journal of Mathematics and Mathematical Sciences

Similarity:

On infinite uniquely partitionable graphs and graph properties of finite character

Jozef Bucko, Peter Mihók (2009)

Discussiones Mathematicae Graph Theory

Similarity:

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property is of finite character if a graph G has a property if and only if every finite induced subgraph of G has a property . Let ₁,₂,...,ₙ be graph properties of finite character, a graph G is said to be (uniquely) (₁, ₂, ...,ₙ)-partitionable if there is an (exactly one) partition V₁, V₂, ..., Vₙ of V(G) such that $G [V_{i}] \in_{i}$ for i = 1,2,...,n. Let us denote by ℜ = ₁ ∘ ₂ ∘ ... ∘ ₙ the class...

Bipartition Polynomials, the Ising Model, and Domination in Graphs

Markus Dod, Tomer Kotek, James Preen, Peter Tittmann (2015)

Discussiones Mathematicae Graph Theory

Similarity:

This paper introduces a trivariate graph polynomial that is a common generalization of the domination polynomial, the Ising polynomial, the matching polynomial, and the cut polynomial of a graph. This new graph polynomial, called the bipartition polynomial, permits a variety of interesting representations, for instance as a sum ranging over all spanning forests. As a consequence, the bipartition polynomial is a powerful tool for proving properties of other graph polynomials and graph...

On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

Similarity:

We deal with the graph operator $\bar{P o w ₂}$ defined to be the complement of the square of a graph: $\bar{P o w ₂} (G) = \bar{P o w ₂ (G)}$ . Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m, n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective...

Difference labelling of cacti

Martin Sonntag (2003)

Discussiones Mathematicae Graph Theory

Similarity:

A graph G is a difference graph iff there exists S ⊂ IN⁺ such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = i,j:i,j ∈ V ∧ |i-j| ∈ V. It is known that trees, cycles, complete graphs, the complete bipartite graphs $K_{n, n}$ and $K_{n, n - 1}$ , pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.

Digraphs with isomorphic underlying and domination graphs: connected $U G^{c} (d)$

Kim A.S. Factor, Larry J. Langley (2007)

Discussiones Mathematicae Graph Theory

Similarity:

The domination graph of a directed graph has an edge between vertices x and y provided either (x,z) or (y,z) is an arc for every vertex z distinct from x and y. We consider directed graphs D for which the domination graph of D is isomorphic to the underlying graph of D. We demonstrate that the complement of the underlying graph must have k connected components isomorphic to complete graphs, paths, or cycles. A complete characterization of directed graphs where k = 1 is presented. ...

A generalization of the dichromatic polynomial of a graph.

Farrell, E.J. (1981)

International Journal of Mathematics and Mathematical Sciences

Similarity:

On graphs with a unique minimum hull set

Gary Chartrand, Ping Zhang (2001)

Discussiones Mathematicae Graph Theory

Similarity:

We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L (v_{i}) = G_{i}$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.

On the signless Laplacian spectral characterization of the line graphs of $T$ -shape trees

Guoping Wang, Guangquan Guo, Li Min (2014)

Czechoslovak Mathematical Journal

Similarity:

A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $D Q S$ ). Let $T (a, b, c)$ denote the $T$ -shape tree obtained by identifying the end vertices of three paths $P_{a + 2}$ , $P_{b + 2}$ and $P_{c + 2}$ . We prove that its all line graphs $ℒ (T (a, b, c))$ except $ℒ (T (t, t, 2 t + 1))$ ( $t \geq 1$ ) are $D Q S$ , and determine the graphs which have the same signless Laplacian spectrum as $ℒ (T (t, t, 2 t + 1))$ . Let $μ_{1} (G)$ be the maximum signless Laplacian eigenvalue of the graph $G$ . We give the limit of $μ_{1} (ℒ (T (a, b, c)))$ , too.

Variations on a sufficient condition for Hamiltonian graphs

Ahmed Ainouche, Serge Lapiquonne (2007)

Discussiones Mathematicae Graph Theory

Similarity:

Given a 2-connected graph G on n vertices, let G* be its partially square graph, obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In particular, this condition is satisfied if x does not center a claw (an induced $K_{1, 3}$ ). Clearly G ⊆ G* ⊆ G², where G² is the square of G. For any independent triple X = x,y,z we define σ̅(X) = d(x) + d(y) + d(z) - |N(x) ∩ N(y) ∩ N(z)|. Flandrin et al. proved that a 2-connected graph G is hamiltonian...

Destroying symmetry by orienting edges: complete graphs and complete bigraphs

Frank Harary, Michael S. Jacobson (2001)

Discussiones Mathematicae Graph Theory

Similarity:

Our purpose is to introduce the concept of determining the smallest number of edges of a graph which can be oriented so that the resulting mixed graph has the trivial automorphism group. We find that this number for complete graphs is related to the number of identity oriented trees. For complete bipartite graphs $K_{s, t}$ , s ≤ t, this number does not always exist. We determine for s ≤ 4 the values of t for which this number does exist.

Rotation and jump distances between graphs

Gary Chartrand, Heather Gavlas, Héctor Hevia, Mark A. Johnson (1997)

Discussiones Mathematicae Graph Theory

Similarity:

A graph H is obtained from a graph G by an edge rotation if G contains three distinct vertices u,v, and w such that uv ∈ E(G), uw ∉ E(G), and H = G-uv+uw. A graph H is obtained from a graph G by an edge jump if G contains four distinct vertices u,v,w, and x such that uv ∈ E(G), wx∉ E(G), and H = G-uv+wx. If a graph H is obtained from a graph G by a sequence of edge jumps, then G is said to be j-transformed into H. It is shown that for every two graphs G and H of the same order (at least...

Histories in path graphs

Ludovít Niepel (2007)

Discussiones Mathematicae Graph Theory

Similarity:

For a given graph G and a positive integer r the r-path graph, $P_{r} (G)$ , has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length r-1, and their union forms either a cycle or a path of length k+1 in G. Let $P_{r}^{k} (G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_{r}^{k} (G)$ . The k-history $P_{r}^{- k} (H)$ is a subgraph of G that is induced by all edges that take part in the recursive...

Uniquely partitionable graphs

Jozef Bucko, Marietjie Frick, Peter Mihók, Roman Vasky (1997)

Discussiones Mathematicae Graph Theory

Similarity:

Let ₁,...,ₙ be properties of graphs. A (₁,...,ₙ)-partition of a graph G is a partition of the vertex set V(G) into subsets V₁, ...,Vₙ such that the subgraph $G [V_{i}]$ induced by $V_{i}$ has property $_{i}$ ; i = 1,...,n. A graph G is said to be uniquely (₁, ...,ₙ)-partitionable if G has exactly one (₁,...,ₙ)-partition. A property is called hereditary if every subgraph of every graph with property also has property . If every graph that is a disjoint union of two graphs that have property also has property...

Contribution to the problem of the spectra of compound graphs.

I. Gutman (1978)

Publications de l'Institut Mathématique [Elektronische Ressource]

Similarity:

Cardinality of a minimal forbidden graph family for reducible additive hereditary graph properties

Ewa Drgas-Burchardt (2009)

Discussiones Mathematicae Graph Theory

Similarity:

An additive hereditary graph property is any class of simple graphs, which is closed under isomorphisms unions and taking subgraphs. Let $L^{a}$ denote a class of all such properties. In the paper, we consider H-reducible over $L^{a}$ properties with H being a fixed graph. The finiteness of the sets of all minimal forbidden graphs is analyzed for such properties.

On Ramsey $(K_{1, 2}, K ₙ)$ -minimal graphs

Mariusz Hałuszczak (2012)

Discussiones Mathematicae Graph Theory

Similarity:

Let F be a graph and let , denote nonempty families of graphs. We write F → (,) if in any 2-coloring of edges of F with red and blue, there is a red subgraph isomorphic to some graph from G or a blue subgraph isomorphic to some graph from H. The graph F without isolated vertices is said to be a (,)-minimal graph if F → (,) and F - e not → (,) for every e ∈ E(F). We present a technique which allows to generate infinite family of (,)-minimal graphs if we know some special graphs. In particular,...