For any positive integer k, let Ak denote the set of finite abelian groups G such that for any subgroup H of G all Cayley sum graphs CayS(H, S) are integral if |S| = k. A finite abelian group G is called Cayley sum integral if for any subgroup H of G all Cayley sum graphs on H are integral. In this paper, the classes A2 and A3 are classified. As an application, we determine all finite Cayley sum integral groups.

Let $G$ be a simple graph. A function $f$ from the set of orientations of $G$ to the set of non-negative integers is called a continuous function on orientations of $G$ if, for any two orientations $O_1$ and $O_2$ of $G$, $|f\left(O_1\right)-f\left(O_2\right)|\le 1$ whenever $O_1$ and $O_2$ differ in the orientation of exactly one edge of $G$. We show that any continuous function on orientations of a simple graph $G$ has the interpolation property as follows: If there are two orientations $O_1$ and $O_2$ of $G$ with $f\left(O_1\right)=p$ and $f\left(O_2\right)=q$, where $p<q$, then for any integer $k$ such that $p<k<q$, there are...

Let $G$ be a graph with order $p$, size $q$ and component number $\omega$. For each $i$ between $p-\omega$ and $q$, let ${𝒞}_i\left(G\right)$ be the family of spanning $i$-edge subgraphs of $G$ with exactly $\omega$ components. For an integer-valued graphical invariant $\varphi$, if $H\to H^{\text{'}}$ is an adjacent edge transformation (AET) implies $|\varphi \left(H\right)-\varphi \left(H^{\text{'}}\right)|\le 1$, then $\varphi$ is said to be continuous with respect to AET. Similarly define the continuity of $\varphi$ with respect to simple edge transformation (SET). Let $M_j\left(\varphi \right)$ and $m_j\left(\varphi \right)$ be the invariants defined by $M_j\left(\varphi \right)\left(H\right)={max}_{T\in {𝒞}_j\left(H\right)}\varphi \left(T\right)$, $m_j\left(\varphi \right)\left(H\right)={min}_{T\in {𝒞}_j\left(H\right)}\varphi \left(T\right)$. It is proved that both $M_{p-\omega }\left(\varphi \right)$ and $m_{p-\omega }\left(\varphi \right)$ interpolate...

In this paper, we consider the intersection graph $I_{\Gamma }\left(\mathbb{Z}ₙ\right)$ of gamma sets in the total graph on ℤₙ. We characterize the values of n for which $I_{\Gamma }\left(\mathbb{Z}ₙ\right)$ is complete, bipartite, cycle, chordal and planar. Further, we prove that $I_{\Gamma }\left(\mathbb{Z}ₙ\right)$ is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of $I_{\Gamma }\left(\mathbb{Z}ₙ\right)$.

In this paper we classify finite groups with disconnected intersection graphs of subgroups. This solves a problem posed by Csákány and Pollák.