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A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. In 1976 Harary and Thomassen proved that the radius $r$ and diameter $d$ of any radially maximal graph satisfy $rle dle 2r-2.$ Dutton, Medidi and Brigham rediscovered this result with a different proof in 1995 and they posed the conjecture that the converse is true, that is, if $r$ and $d$ are positive integers satisfying $rle dle 2r-2,$ then there exists a radially maximal graph with radius $r$ and diameter $d.$ We prove this conjecture and a little more.
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Let $mathscr{G}_{n,beta}$ be the set of graphs of order $n$ with given matching number $beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_{alpha}(G)=alpha D(G)+A(G)$ is called the $alpha$-spectral radius of $G$. The graphs with maximal $alpha$-spectral radius in $mathscr{G}_{n,beta}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $mathscr{G}_{n,beta}$. More precisely, we generalize the known results on the maximal adjacency spectral radius in $mathscr{G}_{n,beta}$ and the signless Laplacian spectral radius.
Let $G$ be a finite simple non-complete connected graph on ${1, ldots, n}$ and $kappa(G) geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $mathrm{diam}(G)$ the diameter of $G$. Being motivated by the computation of the depth of the binomial edge ideal of $G$, the possible sequences $(n, q, f, d)$ of integers for which there is a finite simple non-complete connected graph $G$ on ${1, ldots, n}$ with $q = kappa(G), f = f(G), d = mathrm{diam}(G)$ satisfying $f + d = n + 2 - q$ will be determined. Furthermore, finite simple non-complete connected graphs $G$ on ${1, ldots, n}$ satisfying $f(G) + mathrm{diam}(G) = n + 2 - kappa(G)$ will be classified.
Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleting edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.
Sombor index is a novel topological index introduced by Gutman, defined as $SO(G)=sumlimits_{uvin E(G)}sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}$ denotes the degree of vertex $u$. Recently, Chen et al. [H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022), in press] considered the Sombor indices of trees with given diameter. For the continue, we determine the maximum Sombor indices for unicyclic graphs with given diameter.
We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for Konig-Egervary graphs in terms of their matching numbers.
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