University of Isfahan Transactions on Combinatorics 2251-8657 9 3 2020 09 01 The distance spectrum of two new operations of graphs 125 138 24467 10.22108/toc.2020.116372.1634 EN Zikai Tang Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta- tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Renfang Wu Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta- tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Hanlin Chen Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta- tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Hanyuan Deng Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta- tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China Journal Article 2019 04 09 Let \$G\$ be a connected graph with vertex set \$V(G)={v_1, v_2,ldots,v_n}\$‎. ‎The distance matrix \$D=D(G)\$ of \$G\$ is defined so that its \$(i,j)\$-entry is equal to the distance \$d_G(v_i,v_j)\$ between the vertices \$v_i\$ and \$v_j\$ of \$G\$‎. ‎The eigenvalues \${mu_1, mu_2,ldots,mu_n}\$ of \$D(G)\$ are the \$D\$-eigenvalues of \$G\$ and form the distance spectrum or the \$D\$-spectrum of \$G\$‎, ‎denoted by \$Spec_D(G)\$‎. ‎In this paper‎, ‎we introduce two new operations \$G_1blacksquare_k G_2\$ and \$G_1blacklozenge_k G_2\$ on graphs \$G_1\$ and \$G_2\$‎, ‎and describe the distance spectra of \$G_1blacksquare_k G_2\$ and \$G_1blacklozenge_k G_2\$ of regular graphs \$G_1\$ and \$G_2 \$ in terms of their adjacency spectra‎. ‎By using these results‎, ‎we obtain some new integral adjacency spectrum graphs‎, ‎integral distance spectrum graphs and a number of families of sets of noncospectral graphs with equal distance energy‎. http://toc.ui.ac.ir/article_24467_d2bacaa1da976e615698f1efa93b6605.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 3 2020 09 01 On clique values identities and mantel-type theorems 139 146 24634 10.22108/toc.2020.119553.1680 EN Hossein Teimoori Faal Department of Mathematics and Computer Science, Allameh Tabatabai University, Tehran, Iran. Journal Article 2019 10 08 ‎In this paper‎, ‎we first extend the weighted handshaking‎ ‎lemma‎, ‎using a generalization of the concept of the degree of vertices to the values of graphs‎. ‎This edge-version of the weighted handshaking lemma yields an immediate generalization of the‎ ‎Mantel's classical result which asks for the maximum number of edges in triangle-free graphs‎ ‎to the class of \$K_{4}\$-free graphs‎. ‎Then‎, ‎by defining the concept of value‎ ‎for cliques (complete subgraphs) of higher orders‎, ‎we also‎ ‎extend the classical result of Mantel for any graph \$G\$‎. ‎We finally conclude our paper with a discussion‎ ‎about the possible future works‎. http://toc.ui.ac.ir/article_24634_9320334e5b81d3a3208117a9faddb211.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 3 2020 09 01 On quadrilaterals in the suborbital graphs of the normalizer 147 159 24454 10.22108/toc.2020.120019.1685 EN Serkan Kader Department of Mathematics, Faculty of Arts and Sciences, Niğde &Ouml;mer Halisdemir University, Niğde, Turkey https://orcid.org/0000-0001-5482-5727 Bahadır Ozgur Guler Department of Mathematics, Karadeniz Technical University, Trabzon, Turkey Elif Akşit Department of Mathematics, Niğde &Ouml;mer Halisdemir University, Niğde, Turkey Journal Article 2019 11 11 n this paper‎, ‎we investigate suborbital graphs formed by \$Nbig(Gamma_0(N)big)\$-invariant equivalence relation induced on \$hat{mathbb{Q}}\$‎. ‎Conditions for being an edge are obtained as a main tool‎, ‎then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated‎. http://toc.ui.ac.ir/article_24454_4120f5466b20fdf04f01aaa231ece989.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 3 2020 09 01 Hosoya index of tree structures 161 169 24663 10.22108/toc.2020.121874.1713 EN Ramin Kazemi Department of Statistics, Imam Khomeini International University, Qazvin, Iran Ali Behtoei Department of Pure Mathematics, Imam Khomeini International University, Qazvin, Iran Journal Article 2020 02 28 ‎‎‎The Hosoya index‎, ‎also known as the \$Z\$ index‎, ‎of a graph is the‎ ‎total number of matchings in it‎. ‎In this paper‎, ‎we study the Hosoya index of the tree structures‎. ‎Our aim is to give some results on \$Z\$ in terms of Fibonacci numbers‎ ‎in such structures‎. ‎Also‎, ‎the asymptotic normality of this index is given‎. http://toc.ui.ac.ir/article_24663_a7db688468c0b041742a6b5024dfadb2.pdf
University of Isfahan Transactions on Combinatorics 2251-8657 9 3 2020 09 01 Exact bounds for (λ,n)–stable 0-1 matrices. 171 180 24585 10.22108/toc.2020.120320.1692 EN Trevor Bruen Faculté de Médecine et des sciences de la santé, Université de Sherbrooke, Sherbrooke, Canada Journal Article 2019 12 16 Consider a v × v (0, 1) matrix A with exactly k ones in each row and each column. A is (λ, n)–stable, if it does not contain any λ × n submatrix with exactly one 0. If A is (λ, n)–stable, λ, n ≥ 2, then under suitable conditions on A, v ≥ k k(n−1)+(λ−2) . The case n λ−2 of equality leads to new and substantive connections with block designs. The previous bound and characterization of (λ, 2)–stable matrices follows immediately as a special case. http://toc.ui.ac.ir/article_24585_b98c4f1e8d5aa62e2d43e35851d93e95.pdf