4 Projective planar graphs can be characterized by a set A of 35 excluded minors. However, 5 these 35 are not equally important. A set E of 3-connected members of A is excludable if there 6 are only finitely many 3-connected non-projective planar graphs that do not contain any graph 7 in E as a minor. In this paper we show that there are precisely two minimal excludable sets, 8 which have sizes...

In this paper, we determine the unique graph whose least signless Laplacian eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with maximum degree Δ and among all non-bipartite connected graphs of order n with maximum degree Δ, respectively.

An old conjecture by Jünger, Reinelt and Pulleyblank states that every 2-edgeconnected planar graph can be decomposed into paths of length 3 and triangles, provided its size is divisible by 3. We prove the conjecture for a class of planar graphs including all 2-edge-connected series-parallel graphs. We also present a 2edge-connected non-planar graph that can be embedded on the torus and admits ...

Let G be a non-trivial group, S ⊆ G \ {1} and S = S−1 := {s−1 | s ∈ S}. The Cayley graph of G denoted by Γ(S : G) is a graph with vertex set G and two vertices a and b are adjacent if ab−1 ∈ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integra...

The concept of the bipartite divisor graph for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95–105.]. In this paper, we will consider this graph for the set of character degrees of a finite group G and obtain some properties of this graph. We show that if G is a solvable group, then the number...

We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph G non-isomorphic with G. Let D(G) be the minimal quantifier rank of a such formula. We prove that, if G is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then D(G) = O(log n), where n denotes the order of G. This bound is optimal up to a constant factor. If ...

We describe the structure of 2-connected non-planar toroidal graphs with no K3,3-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K5-subdivision in [A. Gagarin and W. Kocay, “Embedding graphs containing K5-subdivisions”...

In the 1960’s, Tutte presented a decomposition of a 2-connected finite graph into 3-connected graphs, cycles and bonds. This decomposition has been used to reduce problems on 2-connected graphs to problems on 3-connected graphs. Motivated by a problem concerning accumulation points of infinite planar graphs, we generalize Tutte’s decomposition to include all infinite 2-connected graphs.

a r t i c l e i n f o a b s t r a c t Keywords: The Laplace operator for graphs The Harnack inequalities Eigenvalues Diameter We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.