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The diameter of a graph is the maximum distance among all pairs of vertices. Thus a graph $G$ has diameter $d$ if any two vertices are at distance at most $d$ and there are two vertices at distance $d$. We are interested in studying the diameter of circulant graphs $C_n(1,s)$, i.e., graphs with the set $\{0,1,\ldots, n-1\}$ of integers as vertex set and in which two distinct vertices $i,j \in \{0,1,\ldots, n-1\}$ are adjacent if and only if $|i-j|_n\in \{1,s\}$, where $2\leq s\leq \lfloor \frac{n-1}{2} \rfloor$ and $|x|_n=\min(|x|, n-|x|)$. Despite the regularity of circulant graphs, it is difficult to evaluate several parameters, in particular the distance and the diameter. To the best of our knowledge, there is no formulas providing exact values for the distance and the diameter of $C_n(1,s)$ for all $n$ and $s$. In this context, we present in this paper a new approach, based on a simple algorithm, that gives exact values for the distance and the diameter of circulant graphs., Comment: arXiv admin note: text overlap with arXiv:2102.10397

Let $n\geq 5$ and $m\geq 1$ be positive integers. Let $S = ( s_1, s_2, \ldots , s_m)$ be a sequence of integers such that $1 \leq s_1 < s_2 < \ldots < s_m \leq \lfloor \frac{n-1}{2} \rfloor.$ In this paper, we discuss the diameter of multi-loop networks $C_n(s_1,s_2, \ldots, s_m)$ when $s_1=1$. We also present a relation between the diameter of multi-loop networks and the diameter of generalized Petersen graphs., Comment: We proved the existence of exacts values de the diameter that we want to add to this work

In this paper, we study the $L(2, 1)$-Labeling of the Mycielski and the iterated Mycielski of graphs in general. For a graph $G$ and all $t\geq 1$, we give sharp bounds for $\lambda(M^t(G))$ the $L(2, 1)$-labeling number of the $t$-th iterated Mycielski in terms of the number of iterations $t$, the order $n$, the maximum degree $\bigtriangleup$, and $\lambda(G)$ the $L(2, 1)$-labeling number of $G$. For $t=1$, we present necessary and sufficient conditions between the $4$-star matching number of the complement graph and $\lambda(M(G))$ the $L(2, 1)$-labeling number of the Mycielski of a graph, with some applications to special graphs. For all $t\geq 2$, we prove that for any graph $G$ of order $n$, we have $2^{t-1}(n+2)-2\leq \lambda(M^t(G))\leq 2^{t}(n+1)-2$. Thereafter, we characterize the graphs achieving the upper bound $2^t(n+1)-2$, then by using the Marriage Theorem and Tutte's characterization of graphs with a perfect $2$-matching, we characterize all graphs without isolated vertices achieving the lower bound $2^{t-1}(n+2)-2$. We determine the $L(2, 1)$-labeling number for the Mycielski and the iterated Mycielski of some graph classes., Comment: 18 pages, 8 figures

Due to their broad application to different fields of theory and practice, generalized Petersen graphs $GPG(n,s)$ have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for the diameter is still a difficult problem. In their paper, Beenker and Van Lint have proved that if the circulant graph $C_n(1,s)$ has diameter $d$, then $GPG(n,s)$ has diameter at least $d+1$ and at most $d+2$. In this paper, we provide necessary and sufficient conditions so that the diameter of $GPG(n,s)$ is equal to $d+1,$ and sufficient conditions so that the diameter of $GPG(n,s)$ is equal to $d+2.$ Afterwards, we give exact values for the diameter of $GPG(n,s)$ for almost all cases of $n$ and $s.$ Furthermore, we show that there exists an algorithm computing the diameter of generalized Petersen graphs with running time $O$(log$n$)., Comment: We found much general results we want to add to this work