Spectra of toroidal graphs

An n-fold periodic locally finite graph in the Euclidean n-space may be considered the parent of an infinite class of n-dimensional toroidal finite graphs. An elementary method is developed that allows the characteristic polynomials of these graphs to be factored, in a uniform manner, into smaller polynomials, all of the same size.Applied to the hexagonal tessellation of the plane (the graphite sheet), this method enables the spectra and corresponding orthonormal eigenvector systems for all toroidal fullerenes and (3, 6)-cages to be explicitly calculated. In particular, a conjecture of P.W. Fowler on the spectra of (3, 6)-cages is proved.     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

On the role of hypercubes in the resonance graphs of benzenoid graphs

The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent if their symmetric difference forms the edge set of a hexagon of B. A family P of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if B-P contains at least one perfect matching, or if B-P is empty. It is proven that there exists a surjective map f from the set of hypercubes of R(B) onto the resonant sets of B such that a k-dimensional hypercube is mapped into a resonant set of cardinality k.     

Pell graphs

In this paper, we introduce the Pell graphs, a new family of graphs similar to the Fibonacci cubes. They are defined on certain ternary strings (Pell strings) and turn out to be subgraphs of Fibonacci cubes of odd index. Moreover, as well as ordinary hypercubes and Fibonacci cubes, Pell graphs have several interesting structural and enumerative properties. Here, we determine some of them. Specifically, we obtain a canonical decomposition giving a recursive structure, some basic properties (bipartiteness and existence of maximal matchings), some metric properties (radius, diameter, center, periphery, medianicity), some properties on subhypercubes (cube coefficients and polynomials, cube indices, decomposition in subhypercubes), and, finally, the distribution of the degrees.     

Geometry of tropical moduli spaces and linkage of graphs

We prove the following “linkage” theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.     

Spectra of strongly Deza graphs

A Deza graph G with parameters $(n,k,b,a)$ is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children $G_A$ and $G_B$ of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in $G_A$ or $G_B$ if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular.     

Boxicity of line graphs

The boxicity of a graph H, denoted by , is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper we show that for a line graph G of a multigraph, , where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, . For the d-dimensional hypercube Q_d, we prove that . The question of finding a nontrivial lower bound for was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800].The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).     

The face pair of planar graphs

Harary and Kovacs [Smallest graphs with given girth pair, Caribbean J. Math. 1 (1982) 24-26] have introduced a generalization of the standard cage question—r-regular graphs with given odd and even girth pair. The pair (ω,ε) is the girth pair of graph G if the shortest odd and even cycles of G have lengths ω and ε, respectively, and denote the number of vertices in the (r,ω,ε)-cage by f(r,ω,ε). Campbell [On the face pair of cubic planar graph, Utilitas Math. 48 (1995) 145-153] looks only at planar graphs and considers odd and even faces rather than odd and even cycles. He has shown that f(3,ω,4)=2ω and the bounds for the left cases. In this paper, we show the values of f(r,ω,ε) for the left cases where (r,ω,ε)∈{(3,3,ε),(4,3,ε),(5,3,ε), (3,5,ε)}.     

Integral circulant graphs

In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2^τ(n)-1 non-isomorphic integral circulant graphs on n vertices, where τ(n) is the number of divisors of n.     

Eigenvalues and edge-connectivity of regular graphs

In this paper, we show that if the second largest eigenvalue of a d-regular graph is less than , then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x³-(d-3)x²-(3d-2)x-2=0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if the second largest eigenvalue of G is less than , then G is 3-edge-connected.     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

正则图的变换图的谱

设G是一个图,类似全图的定义,可以定义G的8种变换图.如果G是正则图,那么图G的变换图的谱都可以由图G的谱计算得到.