We show that if r s 2; n > r8; and G is a graph of order n containing as many r-cliques as the r-partite Turán graph of order n; then G has more than nr 1=r2r+12 cliques sharing a common edge unless G is isomorphic to the r-partite Turán graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.

In this paper, we established a connection between the Laplacian eigenvalues of a signed graph and those of a mixed graph, gave a new upper bound for the largest Laplacian eigenvalue of a signed graph and characterized the extremal graph whose largest Laplacian eigenvalue achieved the upper bound. In addition, an example showed that the upper bound is the best in known upper bounds for some cases.

For i = 1, . . . . n let C(x;, r,) be a circle in the plane with centre .x i and radius r; . A repeated distance graph is a directed graph whose vertices are the centres and where (xi, x;) is a directed edge whenever x; lies on the circle with centre x, . Special cases are the nearest neighbour graph, when ri is the minimum distance between x, and any other centre, and the furthest neighbour gr...

The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blowups of paths. We determine the extremal number and find the extremal graphs ...

Let Gi be the (unique) 3-graph with 4 vertices and i edges. Razborov [On 3Hypergraphs with Forbidden 4-Vertex Configurations, SIAM J. Discr. Math. 24 (2010), 946–963] determined asymptotically the minimum size of a 3-graph on n vertices having neither G0 nor G3 as an induced subgraph. Here we obtain the corresponding stability result, determine the extremal function exactly, and describe all ex...

A symmetric subset of the reals is one that remains invariant under some reflection x 7→ c − x. We consider, for any 0 < ε ≤ 1, the largest real number ∆(ε) such that every subset of [0, 1] with measure greater than ε contains a symmetric subset with measure ∆(ε). In this paper we establish upper and lower bounds for ∆(ε) of the same order of magnitude: for example, we prove that ∆(ε) = 2ε − 1 ...

Eccentric connectivity index has been found to have a low degeneracy and hence a significant potential of predicting biological activity of certain classes of chemical compounds. We present here explicit formulas for eccentric connectivity index of various families of graphs. We also show that the eccentric connectivity index grows at most polynomially with the number of vertices and determine ...

Institute for Bioinformatics and Translational Research, UMIT, Eduard Wallnoefer Zentrum 1, 6060 Hall in Tyrol, Austria Email: {matthias.dehmer, veronika.kraus}@umit.at (Received May 2, 2012) Abstract We study extremal properties of graph entropies based on so-called information functionals. We obtain some extremality results for the resulting graph entropies which rely on the well-known Shanno...

In this paper, we construct new families of graphs whose automorphism groups are transitive on 3-paths. These graphs are constructed from certain Lie algebras related to generalized Kac-Moody algebras of rank two. We will show that one particular subfamily gives new lower bounds on the number of edges in extremal graphs with no cycles of length fourteen.

A k-uniform hypergraph is s-almost intersecting if every edge is disjoint from exactly s other edges. Gerbner, Lemons, Palmer, Patkós and Szécsi conjectured that for every k, and s > s0(k), every k-uniform s-almost intersecting hypergraph has at most (s + 1) ( 2k−2 k−1 ) edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related result...