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norin results on divisors and jacobians matrix theoretic methods establishes the fundamental matrix characterization a morphism is harmonic iff there exists a diagonal multiplicity matrix satisfying specific adjacency matrix identities proves equivalence with an analogous laplacian matrix identity and an analogous incidence matrix criteria the riemann hurwitz formula presents the graph theoretic analogue relating genera of graphs via harmonic morphisms with matrix proof and applications to regular graphs some functorial consequences demonstrates how harmonic morphisms interact with graph constructions like subdivision edge substitution leaf addition and deletion shows these operations preserve harmonicity under appropriate conditions the chapter ends with exercises and a chapter summary all harmonic morphisms from this graph to c4 are covers fundamental problem given a graph gamma1 for which graphs gamma2 is there a non trivial harmonic morphism phi from gamma2 to gamma1 follow up question can the number of such phi be counted chapter 3 on counting problems this chapter looks at various families such as the path graphs what is especially remarkable is that as we will see the problem of counting harmonic morphisms often boils down to solving certain recurrance relations some of which arose in a completely different context of course in the work of medieval mathematicians both in europe and in india regarding harmonic morphisms between path graphs we show how to construct and count the harmonic morphisms from longer path graphs to shorter ones regarding harmonic morphisms between cycle graphs we show how to construct and count the harmonic morphisms from larger cycle graphs when they exist to smaller ones it turns out all such harmonic morphisms are necessarily covers regarding harmonic morphisms between complete graphs we show how to construct and count the harmonic morphisms from larger complete graphs when they exist to smaller ones harmonic morphisms to p2 arising from the baker norin theorem can be counted harmonic morphisms to p3 the path graph with only 3 vertices can be counted in special cases there are lots of open questions such as which trees have a harmonic morphism to p3 the chapter ends with exercises and a chapter summary chapter 4 on harmonic quotient morphisms this chapter studies quotient graphs arising from group actions and from vertex partitions quotient graphs from group actions harmonic actions and transitive actions are studied separately quotient graphs from paritions orbit partitions and equitable partitions are studied as a nice application of harmonic morphisms with particularly nice structural properties we consider multicovers and blowup graphs the last section provides explicit formulas for the eigenvalue spectra of harmonic blowups of bipartite graphs connecting the eigenvalues of the source and target graphs through the blowup parameters the main result is the godsil mckay theorem the chapter ends with exercises and a chapter summary chapter 5 on graph morphisms and graph products this chapter studies graph morphisms associated to tensor products of graphs and lexicographical products of graphs roughly speaking a graph product of gamma1 with gamma2 is a graph gamma3 v3 e3 where v3 v1 x v2 is the cartesian product and there is a rule for the edges e3 based on some conditions on the vertices the graph products considered in this book are the disjunctive cartesian tensor lexicographic and the strong products the most basic questions one wants answered are these is the projection pr1 gamma1 x gamma2 to gamma1 harmonic and is the projection pr2 gamma1 x gamma2 to gamma2 harmonic if they do turn out to be harmonic morphisms we also want to know the vertical and horizontal multiplicities as well if they do not turn out to be harmonic morphisms we also want if possible to establish conditions on the graphs under which the projections are harmonic however we want to not only consider products of graphs but also products of morphisms in this case the most basic question one wants answered is this given harmonic morphisms phi gamma2 to gamma1 and phi gamma2 to gamma1 is the product phi x phi harmonic for example we show that projection morphisms from tensor products are always harmonic with explicit horizontal multiplicity formulas moreover we prove that the tensor product of harmonic morphisms without vertical edges yields a harmonic morphism with horizontal multiplicity matrix given by the kronecker product of the original multiplicity matrices if gamma x gamma is a lexicographical product then the projection onto the first factor pr1 is a harmonic morphism however the projection onto the second factor is not in general we establish a connection between the balanced blowup graph and a lexicographical product one corollary of this connection is that the blowdown graph agrees with the first projection of the product so is a harmonic morphism chapter 6 on more products and constructions this chapter studies graph morphisms associated to cartesian strong disjunctive products of graphs as well as joins and neps graphs for example we show that projection morphisms from cartesian products or from strong products are always harmonic with explicit horizontal multiplicity formulas roughly speaking one of the results states given two m quasi multicovers phi from gamma2 to gamma1 and phi from gamma2 to gamma1 the cartesian product phi x phi is also an m quasi multicover hence harmonic another result roughly speaking states given two harmonic morphisms phi from gamma2 to gamma1 and phi from gamma2 to gamma1 the strong product phi x phi is also harmonic can one classify the graphs for which the disjunctive product projections pr1 or pr2 are graph morphisms for example we show that if phi from gamma2 to gamma1 and phi from gamma2 to gamma1 are graph morphisms then the associated product map from gamma2 x gamma2 to gamma1 x gamma1 where x is the disjunctive product is in general not a graph morphism given two harmonic morphisms phi from gamma2 to gamma1 and phi from gamma2 to gamma1 the join morphism phi wedge phi is harmonic if and only if a certain technical condition is true a theorem due to urakawa states that projection morphisms from a neps graph to one of its factors are always harmonic moreover we give explicit horizontal multiplicity formulas the chapter ends with exercises and a chapter summary computations are supported throughout using sagemath and mathematica the plan is the publish the volume with birkhauser we thank the editors there especially john benedetto for their encouragement and guidance posted in graph theory group theory harmonic morphisms math research sage tagged combinatorics math mathematics research sage science writing 1 comment the mathematician and the pope posted on 2025 08 21 by wdjoyner acknowledgement this could not have been written without the helpful conversations and correspondences with these brilliant scholars edray goins pomoma john stigall howard nathan alexander howard and susan kelly univ wisconsin retired also i thank the librarians at the catholic university of america for their help i m deeply indebted to them for sharing their knowledge about haynes life and work and philosophy but if there are mistakes and i m sure there are they are my responsibility alone this is a non technical i promise introduction to the life and work of euphemia haynes she s a fascinating character known not just for her pure mathematics phd thesis earned at the age of 53 but for her extraordinary devotion to improving mathematics education for everyone as well as her service to many charities especially those related to the catholic church at the time the prestigious papal decoration of honor medal the pro ecclesia et pontifice was the top award for non clergy in particular all women bestowed by the pope that papal recognition was given to haynes by pope john xxiii when she was almost 70 in fact hers was the only pro ecclesia et pontifice medal bestowed by the pope to anyone during his entire tenure this is an introduction to her life s journey upbringing born martha euphemia lofton in washington d c on september 11 1890 euphemia preferred using her middle name her father dr william lofton was a dentist while her mother lavinia was very active in her church and later became an elementary school teacher in the d c school district according to saved correspondence the family lived on 17th street and attended until euphemia was in her 20s st augustine s which was a few blocks away lavinia and euphemia and joseph were part of the church choir for many years indeed lavinia was the organist for the junior choir since euphemia was a baby this familial foundation within the catholic community likely instilled in euphemia her values of service justice and community engagement education lavinia lofton started teaching in the dc public schools in the fall of 1901 when euphemia was 11 as a kindergarten assistant she was permanently appointed teacher in the dc school system a few years later in the summer of 1903 inspired no doubt by her mother euphemia s educational journey was marked by consistent excellence in fact she distinguished herself early graduating as valedictorian from m street high school in 1907 euphemia lofton began her own teaching career in the dc elementary school system in the fall of 1909 she taught there until the summer of 1912 then she left for smith college where she earned an undergraduate mathematics major and psychology minor in 1914 during this time the letters from her fiance harold haynes discuss his plans to visit her as well as keeping her up to date on various family and business matters in dc upon graduation she immediately returned to dc and started teaching in the high school system in the fall of 1914 she taught in the miner normal school and various local high schools such as dunbar until 1930 at that point she began teaching at the miner teachers college later assimilated with others to become the university of the district of columbia as an early indication of her extraordinary administrative talents euphemia quickly founded and established the mathematics department at miner as well marriage in 1917 she married harold appo haynes a teacher like herself the couple had no children he was a childhood friend and based on saved letters between he and euphemia a source of constant encouragement and strength for her harold had a ee degree from the university of pennsylvania in 1910 and later earned a masters in education from the university of chicago in 1930 and a doctorate in education from new york university in 1946 besides euphemia s promotion to teach at the college level another significant event occurred in 1930 in that year euphemia obtained a master s degree in education from the university of chicago with her husband harold master s thesis euphemia s thesis was a significant piece of scholarship in it she discussed test validity and student assessment methods with the goal to trace the evolution of testing in elementary and secondary school mathematics from 1900 to 1930 she focused on the main mathematics subjects arithmetic algebra and plane geometry she surveyed published education literature analyzed actual test instruments in those subjects and summarized reports by educators and administrators on their own test development in the early 1900s teachers gave a large group of students the same problem and compared how they did without a grading key or separation into different skill metrics in her thesis she notes that educators since those early 1900s started moving away from subjective grading to data driven evaluation tests need to focus objectively on specific abilities tests moved from general surveys to fine grained skill analysis indeed by the 1910s 1920s grade school tests measured discrete skills within core mathematics subjects testing evolved as a tool for diagnosis curriculum evaluation and teacher development she also notes the progression toward standardized tests and its use for diagnosing student challenges in fact after retiring from teaching she because president of the dc board of education the first woman to hold that office by that point she had grown to be an extremely gifted administrator with a single minded focus on service to both education and her church the timing of her presidency following the 1954 brown v board decision and amidst ongoing civil rights litigation positioned her to directly implement and enforce desegregation and equity policies indeed it was during her term as president that the track system was eliminated for further details see the discussion of the 1967 hobson v hansen district court case in ksz14 volunteer service here we mention just a few of the many volunteer efforts euphemia haynes generously devoted her time to a common thread throughout her life was her commitment to social justice and her leadership in various catholic organizations she co founded the catholic interracial council of the district of columbia an organization she helped establish to promote racial harmony and justice within the church and broader community these catholic interracial councils which united under a national umbrella in 1959 1960 were founded with the explicit aim of bridging racial divides between white and black populations the d c council for instance played a vital role in coordinating catholic participation in significant civil rights events haynes also held a significant leadership position as the first vice president of the archdiocesan council of catholic women which focused on education social justice and assisting immigrants she later served as president of the local chapter of that council after retiring from teaching in 1959 she didn t stop working but served others through organizations such as the aauw american association of university women the committee of international social welfare the executive committee of the national social welfare assembly the d c health and welfare council the uso the urban league the naacp and the league of women voters to name a few her unwavering commitment to social justice was fueled by the catholic moral imperative to combat racism her involvement with the catholic interracial council and her broader civil rights work exemplify how catholic social teaching provided a moral framework for her and other social activists during the civil rights era the church s evolving stance on racial justice provided a powerful moral authority and a network such as the catholic interracial council and the archdiocesan council of catholic women through which individuals like haynes could actively pursue desegregation and equality her life demonstrates the practical application of religious principles to pressing social issues showcasing how faith communities were critical sites for civil rights organizing and advocacy phd thesis haynes earned her phd from catholic university of ame...