let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

We present an algorithm for the all pairs shortest distance problem on permutation graphs. Given a permutation model for the graph on n vertices, after O(n) preprocessing the algorithm will deliver answers to distance queries in O(1) time. In the EREW PRAM model, preprocessing can be accomplished in O(log n) time with O(n) work. Where the distance between query vertices is k, a path can be deli...

We establish results on NeST graphs (intersection tolerance graphs of neighborhood subtrees of a tree) and several subclasses. In particular, we show the equivalence of proper NeST graphs and unit NeST graphs, the equivalence of 5xed distance NeST graphs and threshold tolerance graphs, and the proper containment of NeST graphs in weakly chordal graphs. The latter two results answer questions of...

a graph that contains a hamiltonian cycle is called a hamiltonian graph. in this paper wecompute the first and the second geometric – arithmetic indices of hamiltonian graphs. thenwe apply our results to obtain some bounds for fullerene.

a connected graph g is said to be neighbourly irregular graph if no two adjacent vertices of g have same degree. in this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of I-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each I-graph I(n, j, k) admits a unit-distance representation in the Euclidean plane. This implies tha...

In this paper, we are giving MATLAB program to find the energy and Wiener index of unit graphs. Our program demonstrates an intrinsic relationship between the elements of ring and structural properties of graphs. The unit graph G(Zn) turns out to be strongly regular when n = 2k. Several new directions for further research are also indicated by means of raising problems.

The clique-width is known to be unbounded in the class of unit interval graphs. In this paper, we show that this is a minimal hereditary class of unbounded clique-width, i.e., in every hereditary subclass of unit interval graphs the clique-width is bounded by a constant.

In this note, a constructive proof is given that the classes of proper interval graphs and unit interval graphs coincide, a result originally established by Fred S. Roberts. Additionally, the proof yields a linear-time and space algorithm to compute a unit interval representation, given a proper interval graph as input.

Motivated by a satellite communications problem, we consider a generalised colouringproblem on unit disk graphs. A colouring is k-improper if no vertex receives the same colouras k+1 of its neighbours. The k-improper chromatic number χ(G) the least number of coloursneeded in a k-improper colouring of a graph G. The main subject of this work is analysingthe complexity of computin...