Spectral radii of graphs with given chromatic number

Spectral radii of graphs with given chromatic number

We consider the set Gn,k of graphs of order n with the chromatic number k ≥ 2. In this note, we prove that in Gn,k the Turán graph Tn,k has the maximal spectral radius; and Pn if k = 2, Cn if k = 3 and n is odd, C1 n−1 if k = 3 and n is even, K (l) k if k ≥ 4 has the minimal spectral radius. Thus we answer a problem raised by Cao [D.S. Cao, Index function of graphs, J. East China Norm. Univ. Sc...

On the oriented chromatic number of graphs with given excess

The excess of a graph G is defined as the minimum number of edges that must be deleted from G in order to get a forest. We prove that every graph with excess at most k has chromatic number at most 1 2(3+ √ 1 + 8k) and that this bound is tight. Moreover, we prove that the oriented chromatic number of any graph with excess k is at most k+3, except for graphs having excess 1 and containing a direc...

Ela on the Spectral Radii of Graphs without given Cycles

Let G be a graph with n vertices and ρ(G) be the spectral radius of its adjacency matrix. Write Cl for the cycle of order l and let gl(n) = max{ρ(G) : |V (G)| = n, neither Cl nor Cl+1 is a subgraph of G}. This paper obtains the exact value of g5(n) with the unique extremal graph.

On the spectral radii of graphs without given cycles

The locating-chromatic number for Halin graphs

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...