All unicyclic graphs of order n with locating-chromatic number n-3

<p class="p1">Characterizing all graphs having a certain locating-chromatic number is not an easy task. In this paper, we are going to pay attention on finding all unicyclic graphs of order <em>n</em> (⩾ 6) and having locating-chromatic number <em>n</em>-3.</p>

Remarks on the Local Irregularity Conjecture

A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by χirr′(G), is the smallest number of colors used by a locally irregular edge coloring of G. The local irregularity conjecture claims that all graphs, except odd-length paths, odd-length cycles and a certain class of cacti are colorable by three colors. As the conjecture is valid for graphs with a large minimum degree and all non-colorable graphs are vertex disjoint cacti, we study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e., χirr′(B)=4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.

General Randić index of unicyclic graphs with given girth and diameter

We study the general Randić index of a graph [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] is the edge set of [Formula: see text] and [Formula: see text] and [Formula: see text] are the degrees of vertices [Formula: see text] and [Formula: see text], respectively. For [Formula: see text], we present lower bounds on the general Randić index for unicyclic graphs of given diameter and girth, and unicyclic graphs of given diameter. Lower bounds on the classical Randić index and the second modified Zagreb index are corollaries of our results on the general Randić index.

On the Reformulated Multiplicative First Zagreb Index of Trees and Unicyclic Graphs

The multiplicative first Zagreb index of a graph H is defined as the product of the squares of the degrees of vertices of H . The line graph of a graph H is denoted by L H and is defined as the graph whose vertex set is the edge set of H where two vertices of L H are adjacent if and only if they are adjacent in H . The multiplicative first Zagreb index of the line graph of a graph H is referred to as the reformulated multiplicative first Zagreb index of H . This paper gives characterization of the unique graph attaining the minimum or maximum value of the reformulated multiplicative first Zagreb index in the class of all (i) trees of a fixed order (ii) connected unicyclic graphs of a fixed order.