Graphs and Combinatorics最新文献

Fiol has characterized quotient-polynomial graphs as precisely the connected graphs whose adjacency matrix generates the adjacency algebra of a symmetric association scheme. We show that a collection of non-negative integer parameters of size (d + frac{d(d-1)}{2}) is adequate for describing symmetric association schemes of class d that are generated by the adjacency matrix of their first non-trivial relation. We use this to generate a database of the corresponding quotient-polynomial graphs that have small valency and up to 6 classes, and among these find new feasible parameter sets for symmetric association schemes with noncyclotomic eigenvalues.

In this note, we complete a classification of ternary extremal four-negacirculant self-dual codes of lengths 40, 44, 48, 52 and 60.

Galinier, Habib, and Paul introduced the reduced clique graph of a chordal graph G. The nodes of the reduced clique graph are the maximal cliques of G, and two nodes are joined by an edge if and only if they form a non-disjoint separating pair of cliques in G. In this case the weight of the edge is the size of the intersection of the two cliques. A clique tree of G is a tree with the maximal cliques of G as its nodes, where for any (vin V(G)), the subgraph induced by the nodes containing v is connected. Galinier et al. prove that a spanning tree of the reduced clique graph is a clique tree if and only if it has maximum weight, but their proof contains an error. We explain and correct this error. In addition, we initiate a study of the structure of reduced clique graphs by proving that they cannot contain any induced cycle of length five (although they may contain induced cycles of length three or any even integer greater than two). We show that no cycle of length four or more is isomorphic to a reduced clique graph. We prove that the class of clique graphs of chordal graphs is not comparable to the class of reduced clique graphs of chordal graphs by providing examples that are in each of these classes without being in the other.

A q-graph H on n vertices is a set of vectors of length n with all entries from ({0,1,dots ,q}) and every vector (that we call a q-edge) having exactly two non-zero entries. The support of a q-edge ({textbf{x}}) is the pair (S_{textbf{x}}) of indices of non-zero entries. We say that H is an s-copy of an ordinary graph F if (|H|=|E(F)|), F is isomorphic to the graph with edge set ({S_{textbf{x}}:{textbf{x}}in H}), and whenever (vin e,e'in E(F)), the entries with index corresponding to v in the q-edges corresponding to e and (e') sum up to at least s. E.g., the q-edges (1, 3, 0, 0, 0), (0, 1, 0, 0, 3), and (3, 0, 0, 0, 1) form a 4-triangle. The Turán number (mathop {}!textrm{ex}(n,F,q,s)) is the maximum number of q-edges that a q-graph H on n vertices can have if it does not contain any s-copies of F. In the present paper, we determine the asymptotics of (mathop {}!textrm{ex}(n,F,q,q+1)) for many graphs F.

A vertex set in a graph is a connected set if it induces a connected subgraph. For a tree T, each subgraph induced by a connected set of T is actually a subtree of T. The number and average size of subtrees of a tree T are two well-studied parameters. Yan and Yeh developed a linear-time algorithm for computing the number of subtrees in a tree through “generating function”. In this paper, we present linear-time algorithms for computing the number and average size of connected sets in a planar 3-tree.

Chordal graphs are important in structural graph theory. Chordal digraphs are a digraph analogue of chordal graphs and have been a subject of active studies recently. Unlike chordal graphs, chordal digraphs lack many structural properties such as forbidden subdigraph or representation characterizations. In this paper we introduce the notion of semi-strict chordal digraphs which form a class strictly between chordal digraphs and chordal graphs. Semi-strict chordal digraphs have rich structural properties. We characterize semi-strict chordal digraphs in terms of knotting graphs, a notion analogous to the one introduced by Gallai for the study of comparability graphs. We also give forbidden subdigraph characterizations of semi-strict chordal digraphs within the classes of locally semicomplete digraphs and weakly quasi-transitive digraphs.

Given two graphs G and H, the general k-colored Gallai–Ramsey number ({text {gr}}_k(G:H)) is defined to be the minimum integer m such that every k-coloring of the complete graph on m vertices contains either a rainbow copy of G or a monochromatic copy of H. Interesting problems arise when one asks how many such rainbow copy of G and monochromatic copy of H must occur. The Gallai–Ramsey multiplicity ({text {GM}}_{k}(G:H)) is defined as the minimum total number of rainbow copy of G and monochromatic copy of H in any exact k-coloring of (K_{{text {gr}}_{k}(G:H)}). In this paper, we give upper and lower bounds for Gallai–Ramsey multiplicity involving some small rainbow subgraphs.

It is known that every hereditary property can be characterized by finitely many minimal obstructions when restricted to either the class of cographs or the class of (P_4)-reducible graphs. In this work, we prove that the same is true when restricted to some other superclasses of cographs, including (P_4)-sparse and (P_4)-extendible graphs (both of which extend (P_4)-reducible graphs). We also present complete lists of (P_4)-sparse and (P_4)-extendible minimal obstructions for polarity, monopolarity, unipolarity, and (s, 1)-polarity, where s is a positive integer. In parallel to the case of (P_4)-reducible graphs, all the (P_4)-sparse minimal obstructions for these hereditary properties are cographs.

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters (16, 9, 4; 2, 4), we establish the existence of a strictly Neumaier graph with parameters (25, 12, 5; 2, 5), and we disprove the existence of strictly Neumaier graphs with parameters (25, 16, 9; 3, 5), (28, 18, 11; 4, 7), (33, 24, 17; 6, 9), (35, 2212; 3, 5), (40, 30, 22; 7, 10) and (55, 34, 18; 3, 5). Our proofs use combinatorial techniques and a novel application of integer programming methods.

Given a graph T and a family of graphs ({mathcal {F}}), the generalized Turán number of ({mathcal {F}}) is the maximum number of copies of T in an ({mathcal {F}})-free graph on n vertices, denoted by (ex(n,T,{mathcal {F}})). A linear forest is a forest whose connected components are all paths and isolated vertices. Let ({mathcal {L}}_{k}) be the family of all linear forests of size k without isolated vertices. In this paper, we obtained the maximum possible number of r-cliques in G, where G is ({mathcal {L}}_{k})-free with minimum degree at least d. Furthermore, we give a stability version of the result. As an application of the stability version of the result, we obtain a clique version of the stability of the Erdős–Gallai Theorem on matchings.