A class of finite groups with complete character degree graphs

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees.     

Connectivity of intersection graphs of finite groups

The intersection graph of a group G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of G, and there is an edge between two distinct vertices H and K if and only if H∩K≠1 where 1 denotes the trivial subgroup of G. In this paper, we classify finite solvable groups whose intersection graphs are not 2-connected and finite nilpotent groups whose intersection graphs are not 3-connected. Our methods are elementary.     

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.     

Optimization problems of the third edge-connectivity of graphs

The third edge-connectivity λ3(G) of a graph G is defined as the minimum cardinality over all sets of edges, if any, whose deletion disconnects G and each component of the resulting graph has at least 3 vertices. An upper bound has been established for λ3(G) whenever λ3(G) is well-defined. This paper first introduces two combinatorial optimization concepts, that is, maximality and superiority, of λ3(G), and then proves the Ore type sufficient conditions for G to be maximally and super third edge-connected. These concepts and results are useful in network reliability analysis.     

Maximum genus, connectivity and minimal degree of graphs

This paper is devoted to the lower bounds on the maximum genus of graphs. A simple statement of our results in this paper can be expressed in the following form:

Let G be a k-edge-connected graph with minimum degree δ, for each positive integer k(3), there exists a non-decreasing function f(δ) such that the maximum genus γ_M(G) of G satisfies the relation γ_M(G)f(δ)β(G), and furthermore that lim_δ→∞f(δ)=1/2, where β(G)=|E(G)|-|V(G)|+1 is the cycle rank of G.

The result shows that lower bounds of the maximum genus of graphs with any given connectivity become larger and larger as their minimum degree increases, and complements recent results of several authors.     

Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G=(V,E), a family W of sets W⊆V of vertices (areas), and a requirement function r:W→Z⁺ (where Z⁺ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W∈W. So far this problem was shown to be NP-hard in the uniform case of r(W)=1 for each W∈W, and polynomially solvable in the uniform case of r(W)=r?2 for each W∈W. In this paper, we show that the problem can be solved in time, even if r(W)?2 holds for each W∈W, where n=|V|, m=|{{u,v}|(u,v)∈E}|, p=|W|, and r^*=max{r(W)∣W∈W}.     

On the edge-connectivity and restricted edge-connectivity of a product of graphs

The product graph G_m*G_p of two given graphs G_m and G_p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters (λ and λ^′, edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G_m, G_p of minimum degrees δ(G_m), δ(G_p), respectively, we show that λ(G_m*G_p) is lower bounded by both δ(G_m)+λ(G_p) and δ(G_p)+λ(G_m), an improvement of what is known for the edge-connectivity of G_m×G_p.     

Mean distance and minimum degree

We prove that in a graph of order n and minimum degree d, the mean distance μ must satisfy . This asymptotically confirms, and improves, a conjecture of the computer program GRAFFITI. The result is close to optimal; examples show that for any d, μ may be larger than n/(d + 1). © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 95–99, 1997     

Connectivity keeping edges in graphs with large minimum degree

The old well-known result of Chartrand, Kaugars and Lick says that every k-connected graph G with minimum degree at least 3k/2 has a vertex v such that G−v is still k-connected. In this paper, we consider a generalization of the above result [G. Chartrand, A. Kaigars, D.R. Lick, Critically n-connected graphs, Proc. Amer. Math. Soc. 32 (1972) 63–68]. We prove the following result:Suppose G is a k-connected graph with minimum degree at least 3k/2+2. Then G has an edge e such that G−V(e) is still k-connected.The bound on the minimum degree is essentially best possible.     

On cyclic edge-connectivity of transitive graphs

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is said to be cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity cλ(G) is the cardinality of a minimum cyclic edge-cut of G. In this paper, we first prove that for any cyclically separable graph G, , where ω(X) is the number of edges with one end in X and the other end in V(G)?X. A cyclically separable graph G with cλ(G)=ζ(G) is said to be cyclically optimal. The main results in this paper are: any connected k-regular vertex-transitive graph with k≥4 and girth at least 5 is cyclically optimal; any connected edge-transitive graph with minimum degree at least 4 and order at least 6 is cyclically optimal.     

Spectral radius of power graphs on certain finite groups

The power graph of a group $G$ is a graph with vertex set $G$ and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper we find both upper and lower bounds for the spectral radius of power graph of cyclic group $C_n$ and characterize the graphs for which these bounds are extremal. Further we compute spectra of power graphs of dihedral group $D_{2n}$ and dicyclic group $Q_{4n}$ partially and give bounds for the spectral radii of these graphs.     

The connectivity and minimum degree of circuit graphs of matroids

Let G be the circuit graph of any connected matroid M with minimum degree 5（G）. It is proved that its connectivity κ（G） ≥2｜E（M） - B（M）｜ - 2. Therefore 5（G） ≥ 2｜E（M） - B（M）｜ - 2 and this bound is the best possible in some sense.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.