In [2] Fan 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 < g ≤ 2d, and (2...

Let EX (ν; {C3, . . . ,Cn}) denote the set of graphs G of order ν that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n + 1 cycle? We prove that the diameter of G is at most n − 1, and present several results concerning the above question: the girth of G is g = n + 1 if (i) ν...

For a fixed integer, the k-Colouring problem is to decide if vertices of graph can be coloured with at most k colours for an integer k, such that no two adjacent are alike. A G H-free does not contain H as induced subgraph. It known all k≥3, NP-complete graphs contains claw or cycle. The case where cycle follows from result even arbitrarily large girth. We examine what extent situation may chan...

In this paper, we present some new lower and upper bounds for the modified Randic index in terms of maximum, minimum degree, girth, algebraic connectivity, diameter and average distance. Also we obtained relations between this index with Harmonic and Atom-bond connectivity indices. Finally, as an application we computed this index for some classes of nano-structures and linear chains.

Let R be a commutative ring with identity and let I be an ideal of R. Let R 1 I be the subring of R×R consisting of the elements (r,r + i) for r ∈ R and i ∈ I. We study the diameter and girth of the zero-divisor graph of the ring R 1 I.

We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.

A graph is superconnected, for short super-κ, if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any r-regular graph of diameter D and odd girth g that if D ≤ g − 2, then the graph is super-κ when g ≥ 5 and a complete graph otherwise.

We relate the number of minimum cuts in a weighted undirected graph with various structural parameters of the graph. In particular, we provide upper bounds for the number of minimum cuts in terms of the radius, diameter, minimum degree, maximum degree, chordality, girth, and some other parameters of the graph.

Abstract In this paper, we give various lower and upper bounds for the energy of graphs in terms several topological indices graphs: first general multiplicative Zagreb index, Randić zeroth-order redefined indices, atom-bond connectivity index. Moreover, obtain new certain graph invariants as diameter, girth, algebraic radius.