Distance-Regular Graphs

For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

/pdf/a-mathematical-bibliography-of-signed-and-gain-graphs-and-3a49f10hs2.pdf

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

http://emis.matem.unam.mx/journals/EJC/Surveys/ds16.pdf

Dynamic Cage Survey

Generation and properties of snarks

/pdf/on-minimal-regular-graphs-of-girth-6-3u7n5e303s.pdf

On (minimal) regular graphs of girth 6.

Families of small regular graphs of girth 5

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph G q from a projective plane PG(2, q), where q is a prime power. Denote by ex(n; C 4) the maximum number of edges of a graph on n vertices and free of squares C 4. We use the constructed graphs G q to obtain lower bounds on the extremal function ex(n; C 4), for some n < q 2 + q + 1. In particular, we construct a C 4-free graph on $${n=q^2 -\sqrt{q}}$$ vertices and $${\frac{1}{2} q(q^2-1)-\frac{1}{2}\sqrt{q} (q-1) }$$ edges, for a square prime power q.

/pdf/adjacency-matrices-of-polarity-graphs-and-of-other-c4-free-4no5d2vs6n.pdf

Adjacency matrices of polarity graphs and of other C4-free graphs of large size

We study snarks whose edges cannot be covered by fewer than five perfect matchings Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hagglund We construct another infinite family, arising from a generalisation in a different direction The proof that this family has the requested property is computer-assisted In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number five and admit a 5-cycle double cover

Treelike snarks

/pdf/graphs-and-digraphs-with-all-2-factors-isomorphic-1bne1wk808.pdf

Marien Abreu

Papers

On (minimal) regular graphs of girth 6.

Families of small regular graphs of girth 5

Adjacency matrices of polarity graphs and of other C4-free graphs of large size

Treelike snarks

Graphs and digraphs with all 2-factors isomorphic