A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

AMS classifications: 05C50, 05E99;graphs;spectral radius;diameter;bound;degree/diameter

Nonexistence of certain cubic graphs with small diameters

AbstractWe consider the maximum number of vertices in a cubic graph with small diameter. We show that a cubic graph of diameter 4 has at most 40 vertices. (The Moore bound is 46 and graphs with 38 vertices are known.) We also consider bipartite cubic graphs of diameter 5, for which the Moore bound is 62. We prove that in this case a graph with 56 vertices found by Bond and Delorme (1988) is optimal

The degree-diameter problem for sparse graph classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases for fixed $k$. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs

On Middle Cube Graphs

We study a family of graphs related to the $n$-cube. The middle cube graph of parameter k is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors)

Eigenvalue interlacing and weight parameters of graphs

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we show how to use interlacing for proving results about some weight parameters and pseudo-regular partitions of a graph. For instance, generalizing a well-known result of Lov\'asz, it is shown that the weight Shannon capacity $\Theta^*$ of a connected graph \G, with $n$ vertices and (adjacency matrix) eigenvalues $\lambda_1>\lambda_2\ge\...\ge \lambda_n$, satisfies \Theta\le \Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} where $\Theta$ is the (standard) Shannon capacity and \vecnu is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. spectrum are derived

Bipartite regular graphs with fixed diameter

For given nonnegative integers k and D, we consider the problem of determining n,(k, D) , the smallest number n for which there exists a k-regular bipartite graph on n vertices with diameter D. We solve the problem for all pairs ( k, D) with D + 2 (mod 4) and D + 3 (mod 4), for all pairs ( k, D) with k even or k prime and D + 3 (mod 4), for all pairs with D I 9 or k I 4, and for a few other pairs. In the remaining cases, we obtain lower and upper bounds for n,,(k, D). 0 7995 John Wdey & Sons, /nc. 1

On bipartite graphs of defect at most 4

We consider the bipartite version of the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 2, find the maximum number Nb({\Delta},D) of vertices in a bipartite graph of maximum degree {\Delta} and diameter D. In this context, the Moore bipartite bound Mb({\Delta},D) represents an upper bound for Nb({\Delta},D). Bipartite graphs of maximum degree {\Delta}, diameter D and order Mb({\Delta},D), called Moore bipartite graphs, have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree {\Delta} \geq 2, diameter D \geq 2 and order Mb({\Delta},D) - \epsilon with small \epsilon > 0, that is, bipartite ({\Delta},D,-\epsilon)-graphs. The parameter \epsilon is called the defect. This paper considers bipartite graphs of defect at most 4, and presents all the known such graphs. Bipartite graphs of defect 2 have been studied in the past; if {\Delta} \geq 3 and D \geq 3, they may only exist for D = 3. However, when \epsilon > 2 bipartite ({\Delta},D,-\epsilon)-graphs represent a wide unexplored area. The main results of the paper include several necessary conditions for the existence of bipartite $(\Delta,d,-4)$-graphs; the complete catalogue of bipartite (3,D,-\epsilon)-graphs with D \geq 2 and 0 \leq \epsilon \leq 4; the complete catalogue of bipartite ({\Delta},D,-\epsilon)-graphs with {\Delta} \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq \epsilon \leq 4; and a non-existence proof of all bipartite ({\Delta},D,-4)-graphs with {\Delta} \geq 3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs of defect 4 for {\Delta} \geq 3 and D \geq 5, and comment on some implications of our results for upper bounds of Nb({\Delta},D).Comment: 25 pages, 14 Postscript figure