The endomorphism type of certain bipartite graphs and a characterization of projective planes

Fan (in Southeast Asian Bull Math 25, 217-221, 2001) determines the endomorphism type of a finite projective plane. In this note we show that Fan's result actually characterizes the class of projective planes among the finite bipartite graphs of diameter three. In fact, this will follow from a generalization of Fan's theorem and its converse to all finite bipartite graphs with diameter d and girth g such that (1) d + 1 < ga parts per thousand currency sign2d, and (2) every pair of adjacent edges is contained in a circuit of length g

Graph homomorphisms and components of quotient graphs

We study how the number $c(X)$ of components of a graph $X$ can be expressed through the number and properties of the components of a quotient graph $X/\sim.$ We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of component equitable partition, exhibiting interesting inclusions among the various classes of considered homomorphisms. As a consequence, we find a procedure for computing $c(X)$ when the projection on the quotient $X/\sim$ is pseudo-covering. That procedure becomes particularly easy to handle when the partition corresponding to $X/\sim$ is an orbit partition.Comment: arXiv admin note: text overlap with arXiv:1502.0296

Полугруппы сильных эндоморфизмов бесконечных графов и гиперграфов

Визначено один клас нескінченних неорiєнтованих графiв, один клас нескінченних n-однорідних гiперграфiв i доведено, що будь-яка напівгрупа всіх сильних єндоморФізмів графiв i гiперграфiв таких класів ізоморфна вінцевому добутку моноїда перетворень i деякої малої категорії. Знайдено критеріальні умови регулярності напівгрупи сильних ендоморфізмів нескінченних n-однорідних гіперграфів.We define a class of infinite undirected graphs and a class of infinite n-regular hypergraphs and prove that any semigroup of all strong endomorphisms of the graphs and hypergraphs from these classes is isomorphic to the wreath product of a transformation monoid and a small category. We establish the criterional conditions for the regularity of the semigroup of strong endomorphisms of infinite n-regular hypergraphs

Quotient graphs for power graphs

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph $\mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(\mathcal{P}_0(G))$ of its components which is particularly illuminative when $G\leq S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}_0(G)$, the proper order graph $\mathcal{O}_0(G)$ and the proper type graph $\mathcal{T}_0(G)$. We show that all those graphs are quotient of $\mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(\mathcal{P}_0(S_n))$ as well as the number of components of $\widetilde{\mathcal{P}}_0(S_n)$, $\mathcal{O}_0(S_n)$ and $\mathcal{T}_0(S_n)$

Graphs with regular monoids

AbstractThis paper is motivated by an open question: which graphs have a regular (endomorphism) monoid? We present an infinite family of graphs, which possess a regular monoid; we also give an approach to construct a nontrivial graph of any order with this property based on a known one, by which the join of two trees with a regular monoid is explicitly described

On monoids of endomorphisms of a cycle graph

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities

Endomorphism spectra of graphs

AbstractIn this paper we give an account of the different ways to define homomorphisms of graphs. This leads to six classes of endomorphisms for each graph, which as sets always form a chain by inclusion. The endomorphism spectrum is defined as a six-tuple containing the cardinalities of these six sets, and the endomorphism type is a number between 0 and 31 indicating which classes coincide. The well-known constructions by Hedrlin and Pultr (1965) and by Hell (1979) of graphs with a prescribed endomorphism monoid always give graphs of endomorphism type 0 mod 2.After the basic definitions in Section 1, we discuss some properties of the endomorphism classes in Section 2. Section 3 contains what is known about existence of certain endomorphism types, Section 4 gives a list of graphs with given endomorphism type, except for some cases where none have been found so far. Finally we formulate some problems connected with concepts presented here