Uniquely Distinguishing Colorable Graphs

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnected graphs. We introduce two families of uniquely distinguishing colorable graphs, namely type 1 and type 2, and show that every disconnected uniquely distinguishing colorable graph is the union of two isomorphic graphs of type 2. We obtain some results on bipartite uniquely distinguishing colorable graphs and show that any uniquely distinguishing $n$-colorable tree with $n \geq 3$ is a star graph. For a connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if $G$ is uniquely distinguishing colorable of type 1. Also, a characterization of all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$

Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each $k$ there is a threshold graph on the same number of nodes and edges whose sum of the $k$ largest Laplacian eigenvalues exceeds that of the $k$ largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the $k$ largest Laplacian eigenvalues