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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.
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A consequence of Ores classic theorem characterizing the maximal graphs with given order and diameter is a determination of the largest such graphs. We give a very short and simple proof of this smaller result, based on a well-known elementary observation.
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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In this paper we study a pair of numerical parameters associated to a graph $G$. One the one hand, one can construct $text{Hom}(K_2, G)$, a space of homomorphisms from a edge $K_2$ into $G$ and study its (topological) connectivity. This approach dates back to the neighborhood complexes introduced by Lovasz in his proof of the Kneser conjecture. In another direction Brightwell and Winkler introduced a graph parameter called the warmth $zeta(G)$ of a graph $G$, based on asymptotic behavior of $d$-branching walks in $G$ and inspired by constructions in statistical physics. Both the warmth of $G$ and the connectivity of $text{Hom}(K_2,G)$ provide lower bounds on the chromatic number of $G$. Here we seek to relate these two constructions, and in particular we provide evidence for the conjecture that the warmth of a graph $G$ is always less than three plus the connectivity of $text{Hom}(K_2, G)$. We succeed in establishing a first nontrivial case of the conjecture, by showing that $zeta(G) leq 3$ if $text{Hom}(K_2,G)$ has an infinite first homology group. We also calculate warmth for a family of `twisted toroidal graphs that are important extremal examples in the context of $text{Hom}$ complexes. Finally we show that $zeta(G) leq n-1$ if a graph $G$ does not have the complete bipartite graph $K_{a,b}$ for $a+b=n$. This provides an analogue for a similar result in the context of $text{Hom}$ complexes.
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