Path graphs

The concept of a line graph is generalized to that of a path graph. The path graph Pk(G) of a graph G is obtained by representing the paths Pk in G by vertices and joining two vertices whenever the corresponding paths Pk in G form a path Pk+1 or a cycle Ck. P3-graphs are characterized and investigated on isomorphism and traversability. Trees and unicyclic graphs with hamiltonian P3-graphs are characterized

On the Reconstruction Conjecture

"Every graph of order three or more is reconstructible." Frank Harary restated one of the most famous unsolved problems in graph theory. In the early 1900's, while one was working on his doctoral dissertation, two mathematicians made a conjecture about the reconstructibility of graphs. This came to be known as the Reconstruction Conjecture or the Kelly-Ulam Conjecture. The conjecture states: Let G and H be graphs with V(G) = {v_1, v_2, ..., v_n}, V(H) = {u_1, u_2, ..., u_n}, n greater than or equal to 3. If G - v_i is isomorphic to H - u_i for all i = 1, ..., n, then G is isomorphic to H. Much progress has been made toward showing that this statement is true for all graphs. This paper will discuss some of that progress, including some of the families of graphs which we know that the conjecture is true. Another big field of interest about the Reconstruction Conjecture is the information that is retained by a graph when we begin looking at its vertex-deleted subgraphs. Many graph theorists believe that this may show us more about the conjecture as a whole. While working on a possible proof to the Reconstruction Conjecture, many mathematicians began to think about different approaches. One approach that was fairly common was to relate the Reconstruction Conjecture to edges of a graph instead of the vertices. People realized that when deleting only one edge of a graph, then logically more information about the original graph would be retained

Computing the Clique-width of Cactus Graphs

Similar to the tree-width (twd), the clique-width (cwd) is an invariant of graphs. A well known relationship between tree-width and clique-width is that cwd(G) ≤ 3 · 2twd(G)−1. It is also known that tree-width of Cactus graphs is 2, therefore the clique-width for those graphs is smaller or equal than 6. In this paper, it is shown that the clique-width of Cactus graphs is smaller or equal to 4 and we present a polynomial time algorithm which computes exactly a 4-expression

Laplacian Spectral Characterization of Signed Sun Graphs

A sun $SG_{n}$ is a graph of order $2n$ consisting of a cycle $C_{n}$, $n\geq 3$, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if $n$ is odd