We find closed form formulas for the Kemeny’s constant and Kirchhoff index cluster G1{G2} of two highly symmetric graphs G1, G2, in terms parameters original graphs. also discuss some necessary conditions a graph to be symmetric.

Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions. For graphs with...

F , H v are considered simple connected graphs on n and m + 1 vertices, is a specified vertex of u 2 … k ∈ . The graph G = [ ] called with pockets, obtained by taking one copy copies then attaching the i th to 1, at In this article, closed-form formulas resistance distance Kirchhoff index in terms

the first extended zeroth-order connectivity index of a graph   g is defined as 0 1/2 1 ( ) ( ) ,   v   v v g      g d      where   v   (g) is the vertex set of g, and v d is the sum of degrees of neighbors of vertex v in g. we give a sharp lower bound for the first extended zeroth-order connectivity index of trees with given numbers of vertices and pendant vertices,...

Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian G,where D(G) are degree diagonal adjacency matrix, respectively. vertex sequence d1 ≥ d2 ≥··· dn let μ1 μ2 μ n−1 &gt;μn = 0 eigenvalues G. invariants, energy (LE), Laplacian-energy-like invariant (LEL) Kirchhoff index (Kf), defined in terms G, as <m:math xmlns:m="http://www.w3.o...