The integrity of a graph G, I(G), is defined by I(G) = minS⊆V (G){|S| + m(G − S)} where m(G− S) is the maximum order of the components of G− S. In general the integrity of r-regular graph is not known [?]. We answer the following question for special regular graphs. For any given two integers p and r such that pr 2 is an integer, is there a r-regular graph, say G ∗, on p vertices having size q ...

Let G (V, E) be a graph. A subset S of V is called a dominating set of G if every vertex in V-S is adjacent to at least one vertex in S. The domination number γ (G) is the minimum cardinality taken over all such dominating sets in G. A subset S of V is said to be a complementary connected dominating set (ccd-set) if S is a dominating set and < V-S > is connected. The chromatic number χ is the m...

‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

We discuss the finite-size scaling of ferromagnetic Ising model on random regular graphs. These graphs are locally tree-like, and in limit large graphs, Bethe approximation gives exact free energy per site. In thermodynamic limit, these show a phase transition. This transition is rounded off for finite verify theory prediction that this rounding described terms variable $[T/T_c -1] S^{1/2}$ (wh...

Spectral graph theory discusses about the algebraic properties of graphs based on spectrum a graph. This article investigated prism The method used in this research is circulant matrix. results showed that P2,s regular degree 3, for s odd and ≥ circulantt with spectrum.

We prove, that every connected graph with s vertices of degree 3 and t vertices of degree at least 4 has a spanning tree with at least 2 5 t + 1 5 s + α leaves, where α ≥ 8 5 . Moreover, α ≥ 2 for all graphs besides three exclusions. All exclusion are regular graphs of degree 4, they are explicitly described in the paper. We present infinite series of graphs, containing only vertices of degrees...