The Randić index R(G) of a graph G is the sum of the weights (d(u)d(v))− 1 2 of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given ...

The harmonic index H(G) of a graph G is defined as the sum of weights 2 d(u) + d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we first present a sharp lower bound on the harmonic index of unicyclic conjugated graphs (unicyclic graphs with a perfect matching). Also a sharp lower bound on the harmonic index of unicyclic graphs is given in terms of the ...

The revised edge-Szeged index of a connected graph $G$ is defined as Sze*(G)=∑e=uv∊E(G)( (mu(e|G)+(m0(e|G)/2)(mv(e|G)+(m0(e|G)/2) ), where mu(e|G), mv(e|G) and m0(e|G) are, respectively, the number of edges of G lying closer to vertex u than to vertex v, the number of ed...

In a connected graph G, the distance between two vertices of G is length shortest path these vertices. The eccentricity vertex u in largest and any other G. total-eccentricity index ?(G) sum eccentricities all this paper, we find extremal trees, unicyclic bicyclic graphs with respect to index. Moreover, conjugated trees

A caterpillar unicyclic graph is a unicyclic graph in which the removal of all pendant vertices makes it a cycle. In this paper, the unique caterpillar unicyclic graph with minimum algebraic connectivity among all caterpillar unicyclic graphs is determined.

‎For a simple graph $G$‎, ‎the signless Laplacian Estrada index is defined as $SLEE(G)=sum^{n}_{i=1}e^{q^{}_i}$‎, ‎where $q^{}_1‎, ‎q^{}_2‎, ‎dots‎, ‎q^{}_n$ are the eigenvalues of the signless Laplacian matrix of $G$‎. ‎In this paper‎, ‎we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$'s and then determine the unique unicyclic graph with maximum $SLEE$ a...

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend th...

‎in this paper we introduce the concept of order difference interval graph $gamma_{odi}(g)$ of a group $g$‎. ‎it is a graph $gamma_{odi}(g)$ with $v(gamma_{odi}(g)) = g$ and two vertices $a$ and $b$ are adjacent in $gamma_{odi}(g)$ if and only if $o(b)-o(a) in [o(a)‎, ‎o(b)]$‎. ‎without loss of generality‎, ‎we assume that $o(a) leq o(b)$‎. ‎in this paper we obtain several properties of $gamma_...

Let ∆(G), ∆ for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree ∆, where ∆ ≥ n+1 2 (resp., ∆ ≥ n 2 +1 and ∆ ≥ n+3 2 ) are determ...

Let ∆(G), ∆ for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree ∆, where ∆ ≥ n+1 2 (resp., ∆ ≥ n 2 +1 and ∆ ≥ n+3 2 ) are determ...