Chemical graphs, as other ones, are regular if all their vertices have the same degree. Otherwise, they are irregular, and it is of interest to measure their irregularity both for descriptive purposes and for QSAR/QSPR studies. Three indices have been proposed in the literature for that purpose: those of Collatz-Sinogowitz, of Albertson, and of Bell's variance of degrees. We study their propert...

We study maximal Kr+1-free graphs G of almost extremal size—typically, e(G) = ex(n,Kr+1) − O(n). We show that any such graph G must have a large amount of ‘symmetry’: in particular, all but very few vertices of G must have twins. (Two vertices u and v are twins if they have the same neighbourhood.) As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extre...

Let qmin(G) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n: This paper gives some results on the following extremal problem: How large can qmin (G) be if G is a graph of order n; with no complete subgraph of order r + 1? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds ...

We determine the number of perfect matchings and forests in a family Tr,3 of triangulated prism graphs. These results show that the extremal number of perfect matchings in a planar graph with n vertices is bounded below by Ω ( 6 √ 7 + √ 37 n) = Ω (1.535) and the extremal number of forests contained in a planar graph with n vertices is bounded below by

In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdière parameter at most m. We also characterize graphs of maximum spectral radius with no H as a minor when H is either Kr or Ks,t. Interestingly, the extremal graphs match those which maximize the number of edges over all graphs with no H as a minor when r and s are ...

Extremal problems in graph theory form a very wide research area. We study the following topics: the metric dimension of circulant graphs, the Wiener index of trees of given diameter, and the degree‐diameter problem for Cayley graphs. All three topics are connected to the study of distances in graphs. We give a short survey on the topics and present several new results.

In this paper, we determine the largest number of maximal independent sets among all connected graphs of order n, which contain at most one cycle. We also characterize those extremal graphs achieving this maximum value. As a consequence, the corresponding results for graphs with at most one cycle but not necessarily connected are also given.

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to deter...

For a simple graph G with chromatic number x(G), the Nordhaus-Gaddum inequalities give upper and lower bounds for z(G)•(G ¢) and z(G)+ x(GC). Based on a characterization by Fink of the extremal graphs G attaining the lower bounds for the product and sum, we characterize the extremal graphs G for which A(G)B(G c) is minimum, where A and B are each of chromatic number, achromatic number and pseud...

Notation. Given a graph, hypergraph Gn, . . . , the upper index always denotes the number of vertices, e(G), v(G) and χ(G) denote the number of edges, vertices and the chromatic number of G respectively. Given a family L of graphs, hypergraphs, ex(n,L) denotes the maximum number of edges (hyperedges) a graph (hypergraph)Gn of order n can have without containing subgraphs (subhypergraphs) from L...