Spectral radius of power graphs on certain finite groups

The power graph of a group $G$ is a graph with vertex set $G$ and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper we find both upper and lower bounds for the spectral radius of power graph of cyclic group $C_n$ and characterize the graphs for which these bounds are extremal. Further we compute spectra of power graphs of dihedral group $D_{2n}$ and dicyclic group $Q_{4n}$ partially and give bounds for the spectral radii of these graphs.     

A way to construct independence equivalent graphs

Let us denote the independence polynomial of a graph by $I_G\left(x\right)$. If $I_G\left(x\right)=I_H\left(x\right)$ implies that $G?H$ then we say $G$ is independence unique. For graph $G$ and $H$ if $I_G\left(x\right)=I_H\left(x\right)$ but $G$ and $H$ are not isomorphic, then we say $G$ and $H$ are independence equivalent. In [7], Brown and Hoshino gave a way to construct independent equivalent graphs for circulant graphs. In this work we give a way to construct the independence equivalent graphs for general simple graphs and obtain some properties of the independence polynomial of paths and cycles.     

Total domination in inflated graphs

The inflation $G_I$ of a graph $G$ is obtained from $G$ by replacing every vertex $x$ of degree $d\left(x\right)$ by a clique $X=K_{d\left(x\right)}$ and each edge $xy$ by an edge between two vertices of the corresponding cliques $X$ and $Y$ of $G_I$ in such a way that the edges of $G_I$ which come from the edges of $G$ form a matching of $G_I$. A set $S$ of vertices in a graph $G$ is a total dominating set, abbreviated TDS, of $G$ if every vertex of $G$ is adjacent to a vertex in $S$. The minimum cardinality of a TDS of $G$ is the total domination number ${\gamma }_t\left(G\right)$ of $G$. In this paper, we investigate total domination in inflated graphs. We provide an upper bound on the total domination number of an inflated graph in terms of its order and matching number. We show that if $G$ is a connected graph of order $n\ge 2$, then ${\gamma }_t\left(G_I\right)\ge 2n/3$, and we characterize the graphs achieving equality in this bound. Further, if we restrict the minimum degree of $G$ to be at least $2$, then we show that ${\gamma }_t\left(G_I\right)\ge n$, with equality if and only if $G$ has a perfect matching. If we increase the minimum degree requirement of $G$ to be at least $3$, then we show ${\gamma }_t\left(G_I\right)\ge n$, with equality if and only if every minimum TDS of $G_I$ is a perfect total dominating set of $G_I$, where a perfect total dominating set is a TDS with the property that every vertex is adjacent to precisely one vertex of the set.     

A note on the list vertex arboricity of toroidal graphs

The vertex arboricity $a\left(G\right)$ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ such that no cycle is monochromatic. The list vertex arboricity $a_l\left(G\right)$ is the list-coloring version of this concept. In this note, we prove that if $G$ is a toroidal graph, then $a_l\left(G\right)\le 4$; and $a_l\left(G\right)=4$ if and only if $G$ contains $K_7$ as an induced subgraph.