Related papers: A Note on a Conjecture about Commuting Graphs
The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is…
In this paper we introduce the circuit diameter of polyhedra, which is always bounded from above by the combinatorial diameter. We consider dual transportation polyhedra defined on general bipartite graphs. For complete $M{\times}N$…
A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of…
We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a…
Let $R$ and $S$ be commutative rings with identity, $f:R\to S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R\bowtie^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$…
In this paper we introduce compressed commuting graph of rings. It can be seen as a compression of the standard commuting graph (with the central elements added) where we identify the vertices that generate the same subring. The compression…
Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for…
One version of the polycirculant conjecture states that every vertex-transitive graph has a semiregular automorphism. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open case.
We find the Ramsey number of a cycle vs. a complete graph when the order of the cycle is at least 4 times as large as the order of the complete graph. This partially confirms a conjecture of Erd\H{o}s, Faudree, Rousseau, and Schelp made in…
Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to…
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…
A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, the…
In 2002, Koh and Tay conjectured that every bridgeless graph of order $n\geq 5$ and size at least ${n\choose 2}-n+5$ has an orientation of diameter two. Later, Cochran, Czabarka, Dankelmann and Sz\'{e}kely proved this conjecture and asked…
Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of…
In this paper we study zero--divisor graphs of rings and semirings. We show that all zero--divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic…
Let $F$ and $G$ be simple finite undirected graphs. A graph $G$ is called $F$-irregular if any two of its distinct vertices belong to different numbers of copies of $F$ in $G$. According to the strong conjecture about $F$-irregular graphs…
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some…
A strict lower bound for the diameter of a symmetric graph is proposed, which is calculable with the order $n$ and other local parameters of the graph such as the degree $k\,(\geq 3)$, even girth $g\,(\geq 4)$, and number of $g$-cycles…
In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We…
Let R be an Artinian ring and G be the compressed zero-divisor graph associated to R. The question of when the clique number of compressed zero-divisor graphs is finite was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S.…