Extremal total distance of graphs of given radius I

Extremal unicyclic graphs with minimal distance spectral radius

The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m 6= 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) \ Cn.

The distance spectral radius of graphs with given number of odd vertices

Extremal Problems for the p-Spectral Radius of Graphs

The p-spectral radius of a graph G of order n is defined for any real number p > 1 as λ (G) = max 2 ∑ {i,j}∈E(G) xixj : x1, . . . , xn ∈ R and |x1| + · · ·+ |xn| = 1  . The most remarkable feature of λ(p) is that it seamlessly joins several other graph parameters, e.g., λ(1) is the Lagrangian, λ(2) is the spectral radius and λ(∞)/2 is the number of edges. This paper presents solutions to so...

On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs

Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...

Extremal Graphs With a Given Number of Perfect Matchings

Let f(n, p) denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that f(n, p) = n2/4 + cp for some constant cp when n is at least some constant np. For p ≤ 6, they also determined cp and np. For fixed p, we show that the extremal graphs for all n are determined by those with O( √ p) vertices. As a corollary, a com...