The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while holding the other two fixed. Here we focus in {\em bipartite Moore graphs\/}, that is, bipartite graphs attaining the optimum order, fixed either the degree/diameter or degree/girth. The fact that there are very few bipartite Moore graphs suggests the relaxation of some of the constraints implied by the bipartite Moore bound. First we deal with {\em local bipartite Moore graphs}. We find in some cases those local bipartite Moore graphs with local girths as close as possible to the local girths given by a bipartite Moore graph. Second, we construct a family of $(q+2)$-bipartite graphs of order $2(q^2+q+5)$ and diameter $3$, for $q$ a power of prime. These graphs attain the record value for $q=9$ and improve the values for $q=11$ and $q=13$.
A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol, and Fàbrega.\\ Besides, we propose some constructions of bipartite biregular graphs with diameter $d$ and large number of vertices $N(r_1,r_2;d)$, together with their spectra. In some cases of diameters $d=3$, $4$, and $5$, the new graphs attaining the Moore bound are unique up to isomorphism.
Let $b(k,\ell,θ)$ be the maximum number of vertices of valency $k$ in a $(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $θ$. We obtain an upper bound for $b(k,\ell,θ)$ for $0 < θ< \sqrt{k-1} + \sqrt{\ell-1}$. This bound is tight when there exists a distance-biregular graph with particular parameters, and we develop the necessary properties of distance-biregular graphs to prove this.
A bipartite biregular $(n,m;g)$-graph $G$ is a bipartite graph of even girth $g$ having the degree set $\{n,m\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An $(n,m;g)$-bipartite biregular cage is a bipartite biregular $(n,m;g)$-graph of minimum order. In their 2019 paper, Filipovski, Ramos-Rivera and Jajcay present lower bounds on the orders of bipartite biregular $(n,m;g)$-graphs, and call the graphs that attain these bounds {\em bipartite biregular Moore cages}. In parallel with the well-known classical results relating the existen…
Let $b(k,θ)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $θ$. In this paper, we obtain a general upper bound for $b(k,θ)$ for any $0\leq θ< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,θ)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $θ$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no bipartite distance-regular graph…
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite. As main results, we show that in this context the Moore-like bound is attained in the case of diameter $k=3$, and that bipartite mixed graphs of diameter $k\ge 4$ do not exist.
We consider the bipartite version of the degree/diameter problem, namely, given natural numbers Δ \geq 2 and D \geq 2, find the maximum number Nb(Δ,D) of vertices in a bipartite graph of maximum degree Δ and diameter D. In this context, the Moore bipartite bound Mb(Δ,D) represents an upper bound for Nb(Δ,D). Bipartite graphs of maximum degree Δ, diameter D and order Mb(Δ,D), called Moore bipartite graphs, have turned out to be very rare. Therefore, it is very interesting to investigate bipartite graphs of maximum degree Δ \geq 2, diameter D \geq 2 and order Mb(Δ,D) - εwith small ε> 0, that is,…
In this paper we give graphs with the largest known order for a given degree $Δ$ and diameter $D$. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the $(Δ, D)$ table for graphs.
We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this context, the bipartite Moore bound $\M^b(d,D)$ represents a general upper bound for $\N^b(d,D)$. Bipartite graphs of order $\M^b(d,D)$ are very rare, and determining $\N^b(d,D)$ still remains an open problem for most $(d,D)$ pairs. This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on bipartite $(d,D,-4)$-graphs (that is, bipartite gr…
A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \geq 3$ and $t \geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $Θ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s \geq 4$ and $t \geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $Ω( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer …
A (k, g) graph is a graph with regular degree k and girth g. The cage problem refers to finding the smallest (k, g) graph. The (3, 14) cage problem is known to be unresolved. In 2002, Exoo found a (3, 14) record graph with order 384. The trivalent cage problem is restricted in this paper to the Hamiltonian bipartite class of trivalent graphs. A parameter called symmetry factor for representing rotational symmetry is introduced in this paper. The general problem of finding a (3, g) Hamiltonian bipartite graph of minimum order is further decomposed into a set of sub-problems for finding (3, g) H…
The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ via studying spectral properties of their adjacency matrices. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E$ the adjacency matrix of a union of vertex-disjoin…
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minim…
A decorated surface S is an oriented surface with punctures and a finite set of marked points on the boundary, such that each boundary component has a marked point. We introduce ideal bipartite graphs on S. Each of them is related to a group G of type A, and gives rise to cluster coordinate systems on certain spaces of G-local systems on S. These coordinate systems generalize the ones assigned to ideal triangulations of S. A bipartite graph on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d CY category with a cluster collection of spherical objects.…
The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $n-M(k,g).$ A $(k,g)$-cage is a $(k,g)$-graph with the fewest possible number of vertices, among all $(k,g)$-graphs. A graph of diameter $d$ is said to be antipodal if, for any vertices $u, v, w$ such that $d(u,v)=d$ and $d(u, w)=d$, it follows that $d(v, w)=d$ or $v=w.$ In [4] Biggs and Ito proved that any $(k,g)$-cage of even girth $g=2d\geq6$ and excess $e\leq k-2$ is a bipartite graph of diameter $d+1.$ In this …
The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming method for obtaining an upper bound on the order of a regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore, we obtain a general upper bound on the order of a regular uniform…
