Sharp upper bounds for the Laplacian graph eigenvalues

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for $2$-regular bipartite graphs and for $m$-matchings with $m\le 4$.

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

Regular graphs with girth at least 5 and small second largest eigenvalue

The rainbow number of matchings in regular bipartite graphs

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2−1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2−23/18k) in k-regular graphs admitting a perfect matching.

We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when the degrees of all the factors except one or two are small. The resulting asymptotic behavior is seen to generalize the number of semiregular graphs in an elegant way. This leads us to conjecture a general formula when the number of factors is vanishing compared to the number of vertices. As a corollary, we find the average number of ways to partition the edges of a random semiregular bipartite graph into spanning semiregular subgraphs in several cases. Our proof of one case uses a switching argument to find the probability that a set of sufficiently sparse semiregular bipartite graphs are edge-disjoint when randomly labeled.

For given graphs G and H, let |Hom(G, H)| denote the set of graph homomorphisms from G to H. We show that for any finite, n-regular, bipartite graph G and any finite graph H (perhaps with loops), |Hom(G, H)| is maximum when G is a disjoint union of K n,n 's. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of |Hom(G, H)| in terms of a simply expressed parameter of H.

We construct new families of low-density parity-check codes, which we call irregular codes. When decoded using belief propagation, our codes can correct more errors than previously known low-density codes. Our improved performance comes from using codes based on irregular random bipartite graphs, based on the work of Luby et al. (1997). Previously studied low-density codes have been derived from regular bipartite graphs. Initial experimental results for our irregular codes suggest that, with improvements, irregular codes may be able to match turbo code performance.

Finite common coverings of pairs of regular graphs

We study parallel \\emph{Load Balancing} protocols for a client-server\ndistributed model defined as follows.\n There is a set $\\sC$ of $n$ clients and a set $\\sS$ of $n$ servers where each\nclient has\n (at most) a constant number $d \\geq 1$ of requests that must be assigned to\nsome server. The client set and the server one are connected to each other via\na fixed bipartite graph: the requests of client $v$ can only be sent to the\nservers in its neighborhood $N(v)$. The goal is to assign every client request\nso as to minimize the maximum load of the servers.\n In this setting, efficient parallel protocols are available only for dense\ntopolgies. In particular, a simple symmetric, non-adaptive protocol achieving\nconstant maximum load has been recently introduced by Becchetti et al\n\\cite{BCNPT18} for regular dense bipartite graphs. The parallel completion time\nis $\\bigO(\\log n)$ and the overall work is $\\bigO(n)$, w.h.p.\n Motivated by proximity constraints arising in some client-server systems, we\ndevise a simple variant of Becchetti et al's protocol \\cite{BCNPT18} and we\nanalyse it over almost-regular bipartite graphs where nodes may have\nneighborhoods of small size. In detail, we prove that, w.h.p., this new version\nhas a cost equivalent to that of Becchetti et al's protocol (in terms of\nmaximum load, completion time, and work complexity, respectively) on every\nalmost-regular bipartite graph with degree $\\Omega(\\log^2n)$.\n Our analysis significantly departs from that in \\cite{BCNPT18} for the\noriginal protocol and requires to cope with non-trivial stochastic-dependence\nissues on the random choices of the algorithmic process which are due to the\nworst-case, sparse topology of the underlying graph.\n

Short paths in [formula omitted]-regular pairs and small diameter decompositions of dense graphs

The well known “real-life examples” of small world graphs, including the graph of binary relation: “two persons on the earth know each other” contains cliques, so they have cycles of order 3 and 4. Some problems of Computer Science require explicit construction of regular algebraic graphs with small diameter but without small cycles. The well known examples here are generalised polygons, which are small world algebraic graphs i.e. graphs with the diameter d≤clog k−1(v), where v is order, k is the degree and c is the independent constant, semiplanes (regular bipartite graphs without cycles of order 4); graphs that can be homomorphically mapped onto the ordinary polygons. The problem of the existence of regular graphs satisfying these conditions with the degree ≥k and the diameter ≥d for each pair k≥3 and d≥3 is addressed in the paper. This problem is positively solved via the explicit construction. Generalised Schubert cells are defined in the spirit of Gelfand-Macpherson theorem for the Grassmanian. Constructed graph, induced on the generalised largest Schubert cells, is isomorphic to the well-known Wenger’s graph. We prove that the family of edge-transitive q-regular Wenger graphs of order 2qn, where integer n≥2 and q is prime power, q≥n, q>2 is a family of small world semiplanes. We observe the applications of some classes of small world graphs without small cycles to Cryptography and Coding Theory.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

The independent set sequence of regular bipartite graphs

Null graphs (respectively, 1-regular graphs) are the only regular graphs with local antimagic chromatic number 1 (respectively, undefined). In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number 3. As a by-product, we also obtained many families of (possibly disconnected or regular) bipartite and tripartite graph with local antimagic chromatic number 3.