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The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and fully combinatorial, in particular, has no-need of any spectral graph theoretical arguments. Not only the formula itself but also intermediate results through the process of our proof describe clear relations between the HT's of simple random walks on $C^2_N$ and the Fibonacci numbers.

In this study, we present two results that relate Tutte polynomials. First, we provide new and complete polynomial invariants for graphs. We note that the number of variables of our polynomials is one. Second, let L_1 and L_2 be two non-isomorphic lattices. We state that L_1 and L_2 are theta series equivalent if those theta series are the same. The problem of identifying theta series equivalent lattices is discussed in Prof.~Conway's book The Sensual (Quadratic) Form with the title "Can You Hear the Shape of a Lattice?" In this study, we present a method to find theta series equivalent lattices using matroids and their Tutte polynomials., 13 pages, 1 figure

A unit disk graph is a c -strip graph if it has a unit disk representation in which all centers of the unit disks lie between two parallel lines at distance c . The classes of c -strip graphs for various c are studied in the literature. For example, the 0-strip graphs are exactly the unit interval graphs, and every 3 / 2 -strip graph is a co-comparability graph. In this paper, we introduce the class of thin strip graphs and study their properties. A graph is a thin strip graph if it is a c -strip graph for every c > 0 . We show that there is no constant t such that the t -strip graphs are exactly the thin strip graphs. We also show that the class of thin strip graphs properly includes the class of mixed unit interval graphs.

In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants., 7 pages

Two n × n ( 0 , 1 ) matrices X , Y are called thin Lehman matrices if they are solutions of the matrix equation XY T = J n + I n , where J n is the n × n matrix of all 1 s and I n is the identity matrix. These matrices are important in the set covering problem, but few examples are known. In this paper, we will introduce the notion of 1 -overlapped factorizations of finite groups which constructs a new class of thin Lehman matrices. Moreover, we will study some structural properties of 1 -overlapped factorizations.

In 1998, C. Lutolf and F. Margot [C. Lutolf, F. Margot, A Catalog of minimally nonideal matrices, Math. Methods Oper. Res. 47 (1998) 221–241] defined the notion of Lehman matrices in order to classify minimally non-ideal clutters. In the paper [H. Shinohara, Thin Lehman Matrices Arising from Finite Groups, Linear Algebra and Appl. to appear], one of the authors defined the notion of 1-overlapped factorizations of finite group which produces an infinite family of Lehman matrices and proved many properties of 1-overlapped factors. In this paper, we characterize circulant thin Lehman matrices with small constant line sum and provide many useful properties of 1-overlapped factors of cyclic groups.

Near and/or 1-overlapped factorizations on cyclic groups play important roles both in perfect graph theory and ideal clutter theory. Such a factorization is Krasner if its construction does not need any modulo operation (i.e. every addition can be thought as the addition of integers). In this paper, we characterize Krasner near-factorizations and 1-overlapped factorizations, which solves a problem posed by S. Szabo and A.D. Sands [9].

以上の実験結果より次のことがわかった.(1)所定の温度に達してから加圧が開始される本実験の加圧方法では,加熱温度の増加とともに必ずしも密度増加しない.密度が最大値を示す加熱温度は加圧力250,500,750kg/cm2の場合で850℃,1000kg/cm2の場合で950℃であった.この加熱温度の増加とともに必ずしも密度増加しない理由は,加圧によって空孔が粉末粒子間から押し出される度合が加熱温度の高いほど必ずしも大きくないためである.(2)密度比が99.9%以上の高密度成形体は,加圧力1000kg/cm2で加熱温度850℃および950℃の場合に,加圧力750kg/cm2で加熱温度850℃の場合に得られた.(3)加熱温度850℃以上の試料の機械的性質の良否は,その密度の大小に密接に関係し,引張強さは加熱温度850℃の場合に最も大きい値が得られた.(4)加熱温度850℃,加圧力750kg/cm2,加圧時間10minの場合に機械的性質の最も優れた成形体が得られ,その引張強さは35.2kg/mm2,伸びは30.2%であった.