37 research outputs found
Hamiltonicity in connected regular graphs
In 1980, Jackson proved that every 2-connected -regular graph with at most
vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
-regular graphs without a Hamiltonian cycle.Comment: 5 page
The average connectivity matrix of a graph
For a graph and for two distinct vertices and , let
be the maximum number of vertex-disjoint paths joining and in . The
average connectivity matrix of an -vertex connected graph , written
, is an matrix whose -entry is
and let be the spectral
radius of . In this paper, we investigate some spectral
properties of the matrix. In particular, we prove that for any -vertex
connected graph , we have , which implies a result of Kim and O \cite{KO} stating
that for any connected graph , we have ,
where and
is the maximum size of a matching in ; equality holds only when
is a complete graph with an odd number of vertices. Also, for bipartite
graphs, we improve the bound, namely , and equality in the bound
holds only when is a complete balanced bipartite graph
Spectral Bounds for the Connectivity of Regular Graphs with Given Order
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a
graph are measures of its connectivity. These eigenvalues can be used to
analyze the robustness, resilience, and synchronizability of networks, and are
related to connectivity attributes such as the vertex- and edge-connectivity,
isoperimetric number, and characteristic path length. In this paper, we present
two upper bounds for the second-largest eigenvalues of regular graphs and
multigraphs of a given order which guarantee a desired vertex- or
edge-connectivity. The given bounds are in terms of the order and degree of the
graphs, and hold with equality for infinite families of graphs. These results
answer a question of Mohar.Comment: 24 page