Xiaoya Zha

Toughness, trees, and walks

A graph is t-tough if the number of components of G\S is at most |S|-t for every cutset S ⊆ V (G). A k-walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1-walk is a Hamilton cycle, and a longstanding conjecture by Chvatal is that every sufficiently tough graph has a 1-walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k-walk. We fill in the gap between k = 1 and k ≥ 3 by showing that, when k = 2, every sufficiently tough (specifically, 4-tough) graph has a 2-walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k-tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k-walk for k ≥ 2.

The Spectral Radius of Graphs on Surfaces

This paper provides new upper bounds on the spectral radius ? (largest eigenvalue of the adjacency matrix) of graphs embeddable on a given compact surface. Our method is to bound the maximum rowsum in a polynomial of the adjacency matrix, using simple consequences of Eulers formula. Let ? denote the Euler genus (the number of crosscaps plus twice the number of handles) of a fixed surface ?. Then (i) for n?3, every n-vertex graph embeddable on ? has ??2+2n+8??6, and (ii) a 4-connected graph with a spherical or 4-representative embedding on ? has ??1+2n+2??3. Result (i) is not sharp, as Guiduli and Hayes have recently proved that the maximum value of ? is 3/2+2n+o(1) as n?∞ for graphs embeddable on a fixed surface. However, (i) is the only known bound that is computable, valid for all n?3, and asymptotic to 2n like the actual maximum value of ?. Result (ii) is sharp for the sphere or plane (?=0), with equality holding if and only if the graph is a “double wheel” 2K1+Cn?2 (+denotes join). For other surfaces we show that (ii) is within O(1/n1/2) of sharpness. We also show that a recent bound on ? by Hong can be deduced by our method.

The nonorientable genus of complete tripartite graphs

In 1976, Stahl and White conjectured that the nonorientable genus of Kl,m.n, where l ≥ m ≥ n, is ⌈(l-2)(m + n - 2)/2⌉. The authors recently showed that the graphs K3,3,3, K4,4,1, and K4,4,1 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is closely related to voltage graphs.

On the minimal energy of unicyclic Hückel molecular graphs possessing Kekulé structures

Let G be any unicyclic Huckel molecular graph with Kekule structures on n vertices where n>=8 is an even number. In [W. Wang, A. Chang, L. Zhang, D. Lu, Unicyclic Huckel molecular graphs with minimal energy, J. Math. Chem. 39 (1) (2006) 231-241], Wang et al. showed that if G satisfies certain conditions, then the energy of G is always greater than the energy of the radialene graph. In this paper we prove that this inequality actually holds under a much weaker condition.

Orientable and Nonorientable Genera for Some Complete Tripartite Graphs

In this paper, we obtain three general reduction formulas to determine the orientable and nonorientable genera for complete tripartite graphs. As corollaries, we (1) reduce the determination of the orientable (nonorientable, respectively) genera of 75 percent (85 percent, respectively) of nonsymmetric (with respect to l,m, and n) Kl,m,n to that of Km,m,n, and (2) determine the orientable and nonorientable genera for several classes of complete tripartite graphs.

On sharp bounds of the zero-order Randic index of certain unicyclic graphs

Abstract Let G be a simple connected graph and t be a given real number. The zero-order general Randic index α t ( G ) of G is defined as ∑ v ∈ V ( G ) d ( v ) t , where d ( v ) denotes the degree of v . In this paper, for any t , we characterize the graphs with the greatest and the smallest α t within two subclasses of connected unicyclic graphs on n vertices, namely, unicyclic graphs with k pendant vertices and unicyclic graphs with a k -cycle.

On certain spanning subgraphs of embeddings with applications to domination

We prove the existence of certain spanning subgraphs of graphs embedded in the torus and the Klein bottle. Matheson and Tarjan proved that a triangulated disc with n vertices can be dominated by a set of no more than n/3 of its vertices and thus, so can any finite graph which triangulates the plane. We use our existence theorems to prove results closely allied to those of Matheson and Tarjan, but for the torus and the Klein bottle.

Isoperimetric Constants of Infinite Plane Graphs

AbstractLet G be an infinite locally finite plane graph with one end and let H be a finite plane subgraph of G . Denote by a(H) the number of finite faces of H and by l(H) the number of the edges of H that are on the boundary of the infinite face or a finite face not in H . Define the isoperimetric constant h (G) to be infHl(H) / a(H) and define the isoperimetric constant h (δ) to be infG h (G) where the infimum is taken over all infinite locally finite plane graphs G having minimum degree δ and exactly one end. We establish the following bounds on h (δ) for δ ≥ 7 :

On the p-factor-criticality of the Klein bottle

{{(\delta -6)(\delta^2 -8 \delta + 15)}\over {(\delta-4)(\delta^2 - 8\delta + 13)}} \le h (\delta) \le \sqrt {{\delta-6}\over {\delta-2}}.