In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of ...

Let G be a graph with n vertices. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we characterize the graphs (resp. bipartite graphs) with pendent vertices and nullity η, where 0 < η ≤ n. Moreover, the minimum (resp. maximum) number of edges for all (connected) graphs with pendent vertices and nullity η are determined, and the extrem...

Let G be a graph with n vertices. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we characterize the graphs (resp. bipartite graphs) with pendent vertices and nullity η, where 0 < η ≤ n. Moreover, the minimum (resp. maximum) number of edges for all (connected) graphs with pendent vertices and nullity η are determined, and the extrem...

We study Abelian sandpiles on graphs of the form G×I, where G is an arbitrary finite connected graph, and I ⊂ Z is a finite interval. We show that for any fixed G with at least two vertices, the stationary measures μI = μG×I have two extremal weak limit points as I ↑ Z. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show ...

For a graph G, let τ(G) be the decycling number of G and c(G) be the number of vertex-disjoint cycles of G. It has been proved that c(G)≤ τ(G)≤ 2c(G) for an outerplanar graph G. An outerplanar graph G is called lower-extremal if τ(G)= c(G) and upper-extremal if τ(G)= 2c(G). In this paper, we provide a necessary and sufficient condition for an outerplanar graph being upper-extremal. On the other...

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds onD(H) for almost all graphsH and proved that for ‘balanced’H random graphs provide extremal examples and determine the extremal function. Examples of ‘unbalanced graphs’ are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) a...

A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar n...

In the traditional areas of graph theory (Ramsey theory, extremal graph theory, random graphs, etc.), graphs are regarded as abstract binary relations. The relevant methods are often incapable of providing satisfactory answers to questions arising in geometric applications. Geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edg...

Let G = (V, E) be a simple graph with vertex set V (G) = {v1, v2, . . . , vn} and edge set E(G). In this paper, first some sharp upper and lower bounds on the largest and least eigenvalues of graphs are given when vertices are removed. Some conjectures in [M. Aouchiche. Comparaison Automatisée d’Invariants en Théorie des Graphes. Ph.D. Thesis, École Polytechnique de Montréal, February 2006.] an...

The paper shows how to use Extremal Optimization in load balancing of distributed applications executed in clusters of multicore processors interconnected by a message passing network. Composed of iterative optimization phases which improve program task placement on processors, the proposed load balancing method discovers dynamically the candidates for migration with the use of an Extremal Opti...