Preface -- 1 Directed and undirected graphs -- 1.1 Formal description of graphs -- 1.2 Connectedness and equivalence relations -- 1.3 Some special graphs -- 1.4 Homomorphisms -- 1.5 Half-, locally, quasi-strong and metric homomorphisms -- 1.6 The factor graph, congruences, and the Homomorphism Theorem -- Factor graphs -- The Homomorphism Theorem -- 1.7 The endomorphism type of a graph -- 1.8 Comments -- 2 Graphs and matrices -- 2.1 Adjacency matrix -- Isomorphic graphs and the adjacency matrix -- Components and the adjacency matrix -- Adjacency list -- 2.2 Incidence matrix -- 2.3 Distances in graphs -- The adjacency matrix and paths -- The adjacency matrix, the distance matrix and circuits -- 2.4 Endomorphisms and commuting graphs -- 2.5 The characteristic polynomial and eigenvalues -- 2.6 Circulant graphs -- 2.7 Eigenvalues and the combinatorial structure -- Cospectral graphs -- Eigenvalues, diameter and regularity -- Automorphisms and eigenvalues -- 2.8 Comments -- 3 Categories and functors -- 3.1 Categories -- Categories with sets and mappings, I -- Constructs, and small and large categories -- Special objects and morphisms -- Categories with sets and mappings, II -- Categories with graphs -- Other categories -- 3.2 Products & Co -- Coproducts -- Products -- Tensor products -- Categories with sets and mappings, III -- 3.3 Functors -- Covariant and contravariant functors -- Composition of functors -- Special functors - examples -- Mor functors -- Properties of functors -- 3.4 Comments -- 4 Binary graph operations -- 4.1 Unions -- The union -- The join -- The edge sum -- 4.2 Products -- The cross product -- The coamalgamated product -- The disjunction of graphs -- 4.3 Tensor products and the product in EGra -- The box product -- The boxcross product -- The complete product -- Synopsis of the results.

Product constructions as functors in one variable -- 4.4 Lexicographic products and the corona -- Lexicographic products -- The corona -- 4.5 Algebraic properties -- 4.6 Mor constructions -- Diamond products -- Left inverses for tensor functors -- Power products -- Left inverses to product functors -- 4.7 Comments -- 5 Line graph and other unary graph operations -- 5.1 Complements, opposite graphs and geometric duals -- 5.2 The line graph -- Determinability of G by LG -- 5.3 Spectra of line graphs -- Which graphs are line graphs? -- 5.4 The total graph -- 5.5 The tree graph -- 5.6 Comments -- 6 Graphs and vector spaces -- 6.1 Vertex space and edge space -- The boundary & Co -- Matrix representation -- 6.2 Cycle spaces, bases & Co -- The cycle space -- The cocycle space -- Orthogonality -- The boundary operator & Co -- 6.3 Application: MacLane's planarity criterion -- 6.4 Homology of graphs -- Exact sequences of vector spaces -- Chain complexes and homology groups of graphs -- 6.5 Application: number of spanning trees -- 6.6 Application: electrical networks -- 6.7 Application: squared rectangles -- 6.8 Application: shortest (longest) paths -- 6.9 Comments -- 7 Graphs, groups and monoids -- 7.1 Groups of a graph -- Edge group -- 7.2 Asymmetric graphs and rigid graphs -- 7.3 Cayley graphs -- 7.4 Frucht-type results -- Frucht's theorem and its generalization for monoids -- 7.5 Graph-theoretic requirements -- Smallest graphs for given groups -- Additional properties of group-realizing graphs -- 7.6 Transformation monoids and permutation groups -- 7.7 Actions on graphs -- Fixed-point-free actions on graphs -- Transitive actions on graphs -- Regular actions -- 7.8 Comments -- 8 The characteristic polynomial of graphs -- 8.1 Eigenvectors of symmetric matrices -- Eigenvalues and connectedness.

Regular graphs and eigenvalues -- 8.2 Interpretation of the coefficients of chapo(G) -- Interpretation of the coefficients for undirected graphs -- 8.3 Spectra of trees -- Recursion formula for trees -- 8.4 The spectral radius of undirected graphs -- Subgraphs -- Upper bounds -- Lower bounds -- 8.5 Spectral determinability -- Spectral uniqueness of Kn and Kp -- q -- 8.6 Eigenvalues and group actions -- Groups, orbits and eigenvalues -- 8.7 Transitive graphs and eigenvalues -- Derogatory graphs -- Graphs with Abelian groups -- 8.8 Comments -- 9 Graphs and monoids -- 9.1 Semigroups -- 9.2 End-regular bipartite graphs -- Regular endomorphisms and retracts -- End-regular and End-orthodox connected bipartite graphs -- 9.3 Locally strong endomorphisms of paths -- Undirected paths -- Directed paths -- Algebraic properties of LEnd -- 9.4 Wreath product of monoids over an act -- 9.5 Structure of the strong monoid -- The canonical strong decomposition of G -- Decomposition of SEnd -- A generalized wreath product with a small category -- Cardinality of SEnd(G) -- 9.6 Some algebraic properties of SEnd -- Regularity and more for TA -- Regularity and more for SEnd(G) -- 9.7 Comments -- 10 Compositions, unretractivities and monoids -- 10.1 Lexicographic products -- 10.2 Unretractivities and lexicographic products -- 10.3 Monoids and lexicographic products -- 10.4 The union and the join -- The sum of monoids -- The sum of endomorphism monoids -- Unretractivities -- 10.5 The box product and the cross product -- Unretractivities -- The product of endomorphism monoids -- 10.6 Comments -- 11 Cayley graphs of semigroups -- 11.1 The Cay functor -- Reflection and preservation of morphisms -- Does Cay produce strong homomorphisms? -- 11.2 Products and equalizers -- Categorical products -- Equalizers -- Other product constructions.

11.3 Cayley graphs of right and left groups -- 11.4 Cayley graphs of strong semilattices of semigroups -- 11.5 Application: strong semilattices of (right or left) groups -- 11.6 Comments -- 12 Vertex transitive Cayley graphs -- 12.1 Aut-vertex transitivity -- 12.2 Application to strong semilattices of right groups -- ColAut(S,C)-vertex transitivity -- Aut(S, C)-vertex transitivity -- 12.3 Application to strong semilattices of left groups -- Application to strong semilattices of groups -- 12.4 End' (S, C)-vertex transitive Cayley graphs -- 12.5 Comments -- 13 Embeddings of Cayley graphs - genus of semigroups -- 13.1 The genus of a group -- 13.2 Toroidal right groups -- 13.3 The genus of (A × Rr) -- Cayley graphs of A × R4 -- Constructions of Cayley graphs for A × R2 and A × R3 -- 13.4 Non-planar Clifford semigroups -- 13.5 Planar Clifford semigroups -- 13.6 Comments -- Bibliography -- Index -- Index of symbols.