University of IsfahanTransactions on Combinatorics2251-86571320120901A note on the total domination supercritical graphs14182910.22108/toc.2012.1829ENAbdollahAlimadadiShahid Beheshti UniversityChangizEslahchiShahid Beheshti UniversityNaderJafari RadShahrood University of TechnologyJournal Article20120713Let $G$ be a connected spanning subgraph of $K_{s,s}$ and let $H$ be the complement of $G$ relative to $K_{s,s}$. The graph $G$ is <em>$k$-supercritical</em> relative to $K_{s,s}$ if $\gamma_t(G)=k$ and $\gamma_t(G+e)=k-2$ for all $e\in E(H)$. The 2002 paper by T.W. Haynes, M. A. Henning and L.C. van der Merwe, ``Total domination supercritical graphs with respect to relative complements" that appeared in Discrete Mathematics, 258 (2002), 361-371, presents a theorem (Theorem 11) to produce $(2k + 2)$-supercritical graphs relative to $K_{2k+1, 2k+1}$ of diameter $5$, for each $k\geq 2$. However, the families of graphs in their proof are not the case. We present a correction of this theorem.https://toc.ui.ac.ir/article_1829_233087645ace589c1ff0904792d8fee7.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571320120901Subgroup intersection graph of finite abelian groups510186410.22108/toc.2012.1864ENT.Tamizh ChelvamManonmaniam Sundaranar UniversityM.SattanathanManonmaniam Sundaranar UniversityJournal Article20120916Let $G$ be a finite group with the identity $e$. The subgroup intersection graph $\Gamma_{SI}(G)$ of $G$ is the graph with vertex set $V(\Gamma_{SI}(G)) = G-e$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma_{SI}(G)$ if and only if $|\left\langle x\right\rangle \cap\left\langle y\right\rangle|>1$, where $\left\langle x\right\rangle $ is the cyclic subgroup of $G$ generated by $x\in G$. In this paper, we obtain a lower bound for the independence number of subgroup intersection graph. We characterize certain classes of subgroup intersection graphs corresponding to finite abelian groups. Finally, we characterize groups whose automorphism group is the same as that of its subgroup intersection graph.https://toc.ui.ac.ir/article_1864_bfb49638f7dc2dacb195f5bcbbe3091f.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571320120901Hamilton-connected properties in cartesian product1119187110.22108/toc.2012.1871ENRushengulHoshurCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, ChinaElkinVumarCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, ChinaJournal Article20120601In this paper, we investigate a problem of finding natural condition to assure the product of two graphs to be hamilton-connected. We present some sufficient and necessary conditions for $G\Box H$ being hamilton-connected when $G$ is a hamilton-connected graph and $H$ is a tree or $G$ is a hamiltonian graph and $H$ is $K_2$.https://toc.ui.ac.ir/article_1871_ecca2020190157c7d415ed40662b73c2.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571320120901The $L(2,1)$-choosability of cycle2138189510.22108/toc.2012.1895ENHaiyingZhouHong Kong Baptist UniversityWai CheeShiuHong Kong Baptist UniversityPeter Che BorLamUnited International CollegeJournal Article20120727For a given graph $G=(V,E)$, let $\mathscr L(G)=\{L(v) : v\in V\}$ be a prescribed list assignment. $G$ is $\mathscr L$-$L(2,1)$-colorable if there exists a vertex labeling $f$ of $G$ such that $f(v)\in L(v)$ for all $v \in V$; $|f(u)-f(v)|\geq 2$ if $d_G(u,v) = 1$; and $|f(u)-f(v)| \geq 1$ if $d_G(u,v)=2$. If $G$ is $\mathscr L$-$L(2,1)$-colorable for every list assignment $\mathscr L$ with $|L(v)|\geq k$ for all $v\in V$, then $G$ is said to be $k$-$L(2,1)$-choosable. In this paper, we prove all cycles are $5$-$L(2,1)$-choosable.https://toc.ui.ac.ir/article_1895_af9697c011ab75520a7ae549e786aa00.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571320120901The common minimal dominating signed graph3946189610.22108/toc.2012.1896ENP.Siva ReddyDept. of Mathematics, Acharya Institute of Technology, Bangalore-560 090, IndiaB.PrashanthDept. of Mathematics, Acharya Institute of Technology, Bangalore-560 090, IndiaJournal Article20120906In this paper, we define the common minimal dominating signed graph of a given signed graph and offer a structural characterization of common minimal dominating signed graphs. In the sequel, we also obtained switching equivalence characterizations: $\overline{S} \sim CMD(S)$ and $CMD(S) \sim N(S)$, where $\overline{S}$, $CMD(S)$ and $N(S)$ are complementary signed graph, common minimal signed graph and neighborhood signed graph of $S$ respectively.https://toc.ui.ac.ir/article_1896_1e6fda2b7401644ed7d16a77256e20b5.pdfUniversity of IsfahanTransactions on Combinatorics2251-86571320120901The eigenvalues and energy of integral circulant graphs4756190910.22108/toc.2012.1909ENMohsenMollahajiaghaeiAmirkabir UniversityJournal Article20121010A graph is called \textit{circulant} if it is a Cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. Let $D$ be a set of positive, proper divisors of the integer $n>1$. The integral circulant graph $ICG_{n}(D)$ has the vertex set $\mathbb{Z}_{n}$ and the edge set E$(ICG_{n}(D))= \{\{a,b\}; gcd(a-b,n)\in D \}$. Let $n=p_{1}p_{2}\cdots p_{k}m$, where $p_{1},p_{2},\cdots,p_{k}$ are distinct prime numbers and $gcd(p_{1}p_{2}\cdots p_{k},m)=1$. The open problem posed in paper [A. Ili\'{c}, The energy of unitary Cayley graphs, Linear Algebra Appl., 431 (2009) 1881--1889] about calculating the energy of an arbitrary integral circulant $ICG_{n}(D)$ is completely solved in this paper, where $D=\{p_{1},p_{2},\ldots,p_{k} \}$.https://toc.ui.ac.ir/article_1909_57978492f5c7801ebbae87453714acbf.pdf